Camille Cany

Christine Mansilla
email: christine.mansilla@cea.fr

Pascal Da Costa

Gilles Mathonnière

Thierry Duquesnoy

Anne Baschwitz

Nuclear power fleet replacement: an opportunity for the French energy mix?

In France, 27% of the electricity is to be produced by renewable resources by 2020. This share is intended to grow continuously up to 2050. The recent European agreement and the French "energy transition law" will promote such a development. The French power system is characterized by high nuclear penetration and nuclear power is meant to remain a significant contributor in the medium and long term, as a low-carbon power source. More than half the French nuclear power fleet was installed in the late seventies / early eighties. Thus, the issue of its replacement is at the core of the French power mix issue. The objective of this paper is to provide some insights about the opportunity it enables for the energy mix. Two plausible replacement scenarios are developed and analyzed as regards the energy cost provided by nuclear power. For a given target level of nuclear installed capacities, the penetration of non-dispatchable renewables with dispatch priority will increase the need for nuclear power modulation at reduced average load factor. The impact of modulation on the nuclear levelized cost of electricity is assessed, according to the considered replacement scenario and for different renewable and nuclear energy penetration scenarios. Results show that, according to the selected assumptions, implementing a progressive shut-down (based on an increased operation lifetime of Nuclear Power Plants) appears a relevant choice since it both provides a lowest power production cost even at reduced average load factor to participate to load following and allows the possibility of "waiting" for choosing most sustainable technologies.

I. INTRODUCTION

The general 3X20 European directive proposes renewable penetration goals 1 . In France, 27% of the electricity is to be produced by renewable resources by 2020 [START_REF] De L'ecologie | Plan d'action national en faveur des énergies renouvelables[END_REF] and this share is intended to continuously grow up to 2050. In 2013, the share of renewable power already reached 18% of the domestic production, namely approximately 100 TWh [START_REF] Rte | Bilan prévisionnel de l'équilibre offre-demande d'électricité en France -édition 2014[END_REF] .

Among these resources, some are not dispatchable (i.e. they cannot be started (or stopped) at the Transmission System Operator request), which triggers challenges to maintain the reliability target level of the power system, both in the short and long term [START_REF] Gross | The costs and impacts of intermittency: an assessment of the evidence on the costs and impacts of intermittent generation on the British electricity network[END_REF][START_REF] Hart | The potential of intermittent renewables to meet electric power demand: current methods and emerging analytical techniques[END_REF] . Wind and solar are expected to contribute to about 10% of the French electricity production in 2020 [START_REF] Rte | Bilan prévisionnel de l'équilibre offre-demande d'électricité en France -édition 2014[END_REF][START_REF] Ancre | Scénarios de l'ANCRE pour la transition énergétique[END_REF] , and according to voluntaristic scenarios they could contribute to over 50% of the total electricity production by 2050 [START_REF] Ademe | L'exercice de prospective de l[END_REF] . The recent European agreement appears voluntaristic; by announcing a binding target of at least 27% of renewable energy used at the European level by 2030, and will promote such a development [START_REF]Energy and climate goals for 2030[END_REF] . The French power system is currently characterized by a high nuclear penetration: it supplied 82% of the French domestic consumption and 73% of the total demand (including exportations) in 2013 [START_REF] Rte | Bilan prévisionnel de l'équilibre offre-demande d'électricité en France -édition 2014[END_REF] . As a matter of fact, nuclear power is meant to remain a significant contributor to the French power system in the medium term, as a lowcarbon power source.

More than half the French nuclear power fleet was installed in the late seventies / early eighties [START_REF]Power Reactor Information System (PRIS) database[END_REF] . Thus, the issue of its replacement is at the core of the French power mix issue: the objective of this paper is to provide some insights about the opportunity it enables for the energy mix.

In an academic approach, first two plausible replacement scenarios are developed and analyzed as regards the energy cost provided by nuclear power, according to the composition of the nuclear mix. Then, different scenarios in terms of non-dispatchable renewable and nuclear energy penetration are confronted in order to identify the availability for a "nuclear back-up", in a lowcarbon power mix perspective. Finally, the impact of modulation (loss of load factor) on the nuclear levelized cost of electricity is assessed, according to the considered replacement scenario.

II. NUCLEAR FLEET REPLACEMENT SCENARIOS

II.A. Scenario Design

In order to investigate the issue of the nuclear fleet replacement with an academic viewpoint, two plausible scenarios have been designed as regards the French operational nuclear fleet. The first one is driven by the power plants' age while the second one is designed to be progressive, in order to smooth the investment requirements:

-Current reactors are operated 40 years and then shut down ("40 yrs" scenario); -The current fleet is progressively replaced to avoid massive investment in a short period of time: 2 GWe are shut down each year starting from 2017 ("prog." scenario). Other more progressive scenarios could also have been considered.

In each case, the current fleet made of Pressurized Water Reactors (PWR) is completed by Evolutionary Power Reactors (EPR™) in order to achieve the target nuclear penetration level of the selected scenario (an example is provided in Fig. 1 for a 50% nuclear penetration in terms of produced energy; the total energy production is assumed to be 543TWh (2012-2013 level) 10 , load factors are assumed to be 0.75 for PWRs and 0.8 for EPRs™). As a result, the installed capacity for each nuclear technology can be assessed according to the selected replacement scenario, as displayed in Fig. 1. In both scenarios, the nuclear contribution to the total energy production of the power mix declines from 73% in 2015 to 50% in 2030, when it stabilizes at this level (for a 50% nuclear penetration case, as illustrated in Fig. 1. Other assumptions will be examined in sections III and IV).

In the "prog." scenario, the PWRs shut down is slower than in the "40 yrs" scenario: for a given year, more PWRs are in operation for the same level of nuclear production capacity.

II.B. Impact on the Nuclear Fleet Production Cost

The scenarios presented in the previous section have been established based on realistic expectations. They will result in different annual levelized production costs according to the composition of the nuclear mix:

the Capex of the nuclear mix is computed from the share between amortized power plants (PWR) and recently installed ones (EPR ™); -Opex and life extension costs (up to 2025) have also been included; -Increased load factor is assumed for EPR™; -A learning rate is assumed for the EPR™, investment cost is decreased from the first-of-akind to the tenth power plant, and then considered constant. Recent cost assumptions are issued from the French Court of Auditors in-depth study of the nuclear costs in the French context [START_REF] Cour Des Comptes | Les coûts de la filière électronucléaire[END_REF] .

The nuclear annual levelized production cost of electricity is computed according to the retained assumptions and the nuclear fleet replacement scenario. To this end, the "Economic Current Cost" methodology has been retained for a discount rate of 7.8% (i.e. annual levelized production cost is assessed by annualizing the investment including interests over the selected amortization period. After this date, the investment burden is zero: this is the characteristic of this method). The results are presented on Fig. 2 for a 50% nuclear penetration in terms of produced energy, between 2030 and 2050: the two deadlines that will be considered in next sections. The ratio between the nuclear annual levelized cost of electricity for the "prog." scenario and the "40 yrs" scenario is displayed in the figure .   In 2030, life extension costs are already taken into account. As a result, they do not appear on the Fig. 2 results. In 2030, there are almost no more PWRs in the fleet for the "40 yrs" scenario (approx. 10 GW), whereas the PWR power is still significant in the "prog." scenario (approx. 40 GW). As a result, from 2030 to 2050, the "prog." nuclear generation cost appears less expensive, first because EPRs™ are introduced at a slower pace than in the "40 yrs" scenario, second because it takes advantage of amortized PWRs. The scenarios converge by 2050, since the nuclear fleet is only composed by EPRs™ by then, for both scenarios.

Thus, results highlight the interest of taking advantage of amortized power plants, as long as additional costs due to extended operating life are not too high.

III. TOWARDS A LOW-CARBON POWER MIX: RENEWABLE AND NUCLEAR PENETRATION PROSPECTIVE SCENARIOS

III.A. Scenario Design

The aim is to achieve a low-carbon power mix in the medium to long term, the so-called energy transition that needs to take into account the national specificities (power mix history and inertia, regional resources etc.).

In this respect, the French Government launched a broad national consultation on the energy transition in 2013 [START_REF] De L'ecologie | Synthèse des travaux du débat national sur la transition énergétique de la France[END_REF] . In this framework, several institutions designed energy prospective scenarios for France. They propose scenarios in terms of renewable and nuclear energy penetration.

The Institute for techno-economics of energy systems of the Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA/I-tésé) was involved in the contribution of the ANCRE (Alliance Nationale de Coordination de la Recherche pour l'Energie, the French National Alliance for Energy Research Coordination), that proposed three contrasted scenarios that all satisfy the factor 4, i.e. a reduction by four of the greenhouse gas emissions in 2050, compared to the 1990 level [START_REF] Ancre | Scénarios de l'ANCRE pour la transition énergétique[END_REF] . The specificity of these scenarios is a voluntaristic approach in terms of scientific and technological innovation. All these scenarios propose a similar non-dispatchable renewable penetration for a given time horizon, leading to a maximum energy penetration of 30% by 2050 (wind and solar). As regards nuclear power, two cases are considered: a 50% value (of the annual electricity production), in line with present status of the "energy transition law", and a "high nuclear" case with a 65% value.

In order to consider an "extreme" scenario in terms of non-dispatchable renewable penetration, we also considered the scenario proposed by the ADEME (Agence de l'Environnement et de la Maîtrise de l'Energie, the French Environment and Energy Management Agency), which examines the case of 50% non-dispatchable renewable penetration by 2050 [START_REF] Ademe | L'exercice de prospective de l[END_REF] . The ADEME also considered lower nuclear penetration (down to 20%).

At the opposite, as regards nuclear power contribution to the French electricity mix, we also considered a "business-as-usual" value of 75%, which corresponds to the nuclear current contribution to the French annual electricity production [START_REF] Rte | Bilan prévisionnel de l'équilibre offre-demande d'électricité en France -édition 2014[END_REF] . II gather the assumptions selected for non-dispatchable renewable resources and nuclear share in the total French power production according to the ANCRE and ADEME scenarios. From them, we established generic scenarios by crossing the renewable and nuclear penetration assumptions, and studied each of them. For readability issues, only three contrasted cases will be presented in the results. They provide conceivable target capacities in line with the current energy policy as regards renewable energy development and the specific French context characterized by the significant place of nuclear power in the power mix:

Table I and Table

1. "median scenario" (med): 16.5% wind and solar in 2030, 30% in 2050, 50% nuclear in both cases; 2. "high nuclear scenario" (high nuc): 65% nuclear, wind and solar shares as in the median case; 3. "high non-dispatchable renewable scenario" (high ren): 30% wind and solar in 2030, 50% in 2050, 50% nuclear in both cases.

In each scenario, we deliberately focus on nuclear and non-dispatchable renewable resources without clarifying the remaining part of the energy production which is off topic here.

III.B. Impact on the Nuclear Fleet Load Factor

Renewable resources have today a dispatch priority: they have a priority access to the grid and the rest of the power system should adapt to renewable energy production. To assess the impacts of non-dispatchable renewable resources on other power plants capacity requirements and annual productions in a power system, two major types of approaches can be distinguished.

First, the construction of load duration curves net of wind and solar (called "residual" in what follows) for different wind and solar penetration levels allows determining the optimal mix of other generation resources in the long term [START_REF] Davis | Determining the impact of wind on system costs via the temporal patterns of load and wind generation[END_REF][START_REF] Keppler | Nuclear energy and renewables: system effects in low-carbon electricity systems[END_REF][START_REF] Ueckerdt | System LCOE: What are the costs of variable renewables?[END_REF] .

Secondly, a minimum cost economic dispatch model can serve refining the general trends obtained from the first approach [START_REF] Davis | Determining the impact of wind on system costs via the temporal patterns of load and wind generation[END_REF][START_REF] Keppler | Nuclear energy and renewables: system effects in low-carbon electricity systems[END_REF][START_REF] Connolly | Modelling the existing Irish energy-system to identify future energy costs and the maximum wind penetration feasible[END_REF][START_REF] Barnacle | Modelling generation and infrastructure requirements for transition pathways[END_REF] . The use of such a time step model allows short term supply and demand balancing in response to technical, economic and political constraints.

We focus here on the first approach to put a light on general trends in the long term.

The developed methodology proceeds in several steps. On the one hand, the construction of annual residual load duration curves allows assessing the annual energy production to be fulfilled by dispatchable power plants. The need for nuclear energy production is then deduced knowing its contribution to the system as a "baseload" power. On the other hand, nuclear load following dedicated energy is evaluated based on the realistic French nuclear fleet scenarios that were presented in the previous section. Thanks to the annual residual load duration curves, we evaluate and compare nuclear fleet effective full power hours.

III.B.1. Construction of Annual Residual Load Duration Curves

A load duration curve represents the sorted hourly load of one year, starting with the highest load hour. The residual load curve is obtained by withdrawing the wind and solar production to the total annual production (considered constant on an annual cumulated basis in this study, hourly variations have been computed into the residual load duration curve based on the historical data from the French Transmission System Operator (RTE)). Wind and solar production was estimated by calculating mean hourly power values of historical data (between 2010 and 2013) also from RTE 10 , and by calibrating their hourly power to fit their assumed energy penetration level to satisfy a given demand (cf. Table I). The resulting curves are displayed on Fig. 3.

Even if wind and solar penetrations are aggregated in the displayed rates, separate assumptions were considered and the corresponding production profiles were treated separately in the calculations.

It should also be noted that the considered demand, to which wind and solar production are withdrawn, actually includes the exportations. Thus, exportations are assumed exogenous. Additional exportations in case of excess capacities for certain hours may be of interest, if the interconnection capacities allow it.

III.B.2. Assessment of Nuclear Modulation Availability

As regards the use of nuclear power, we retained an empirical approach based on the historical availability factor of nuclear in France (return on operating experience) [START_REF]Power Reactor Information System (PRIS) database[END_REF] . The selected value is 7000 hours. This value is used to assess the "baseload" nuclear power (called P baseload in what follows), according to the residual load to supply. The "baseload" nuclear production (corresponding to an installed capacity of P baseload ) will then be compared to the nuclear production corresponding to the installed nuclear power provided by the selected scenario, to assess the available load following available energy (cf. Fig. 4). Firstly, scenarios are compared through the corresponding "baseload" nuclear production. We adapted Fig. 4 to each renewable penetration scenario and calculated the share of the total annual electricity production (i.e. 543TWh) that would be provided by nuclear, according to the renewable penetration, when nuclear is operated at least 7000 hours per year. Results are displayed in Table III. From this table, we can point out that, for 16.5% renewable penetration, 57% of the electricity should be supplied by nuclear as "baseload", according to the retained assumptions. Hence, 50% nuclear penetration (corresponding to the "med" scenario in 2030), would be underestimated compared to the "baseload" reference and thus no nuclear load following capacity would be available.

III.B.3. Assessment of Nuclear Load Factor according to Renewable Penetration

When the nuclear share is higher than what was identified in Table III as the "baseload" power, some load following capacity is available. In other words, the average nuclear load factor is decreased for the total fleet. Table IV presents the results according to the scenarios, as the load factor reduction compared to the "med 2030" case for which there is no modulation ("Ref" case). Nuclear power would then contribute to a higher extent to balancing services. Further R&D works are still needed though, to verify if implementing higher modulation needs appears feasible from both technical and economic viewpoints. The relevance of operating nuclear power plants with low load factors would be indeed highly linked with payments for grid services (i.e. reduction of system costs). Other tracks should also be explored, such as taking advantage from the available energy to produce valuable services and products. Detailed business models are still to be developed in this field.

IV. CONFRONTING THE NUCLEAR FLEET REPLACEMENT SCENARIOS

Considering the nuclear and renewable penetration scenarios that have been defined in section III (med, high nuc, high ren), and the nuclear fleet replacement scenarios that have been presented in section II, we are now confronting the nuclear fleet cost resulting from both the renewable and nuclear energy penetration (impact of load factor reduction), and selected fleet replacement (impact of the nuclear fleet composition). For each nuclear penetration scenario and nuclear fleet replacement policy, the nuclear annual production costs are calculated based on the total nuclear annual production (i.e. the load factor assessment depending on the wind and solar penetration scenario, cf. Fig. 4).

By 2050, both nuclear replacement scenarios lead to a fleet made of EPR™ exclusively, hence costs converge. In what follows, we focus on 2030 to point out differences between scenarios.

In section II, it was shown that in 2030, the "prog." scenario leads to lower nuclear production costs than the "40 yrs" scenario. In section III, it was highlighted that nuclear load following (particularly resulting from renewable dispatch priority) induced reduced average load factors, hence higher production costs.

Therefore, the question arises: is the nuclear "prog." replacement scenario still cheaper than the "40 yrs" scenario, even when operating the nuclear fleet in a load following mode? To answer this question, the comparison is carried out against the "40 yrs" replacement scenario with nuclear operating as "baseload" (cf. section III).

We compare annual levelized production costs in the two following cases:

Case 1: for 16.5% renewable penetration in 2030, "high nuc" scenario (65% nuclear), with a nuclear fleet composition derived from the "prog." scenario, costs are compared to the "baseload" nuclear penetration with a nuclear fleet composition derived from the "40 yrs" scenario. Case 2: for 30% renewable penetration in 2030, "high ren" scenario (50% nuclear), with a nuclear fleet composition derived from the "prog." Scenario, costs are compared to the "baseload" nuclear penetration with a nuclear fleet composition derived from the "40 yrs" scenario.

Relative differences of nuclear annual levelized costs for case 1 and 2 are displayed in Table V. -33 % For 16.5% and 30% non-dispatchable renewable penetration in 2030, the "prog" scenario nuclear replacement policy allows to consider the possibility for nuclear to operate at reduced load factors in order to participate in load following. Indeed, in such case, the annual nuclear levelized production cost still remains lower than the case with nuclear operating as "baseload" with a "40 yrs" replacement policy. It makes it possible to take advantage of the amortized power plants, the history of the French power mix.

Besides enabling a lower production cost for the nuclear fleet, smoothing the replacement of the nuclear capacities would be also beneficial for the whole power mix. Indeed, in the case of which nuclear power would be partly replaced by renewables, delaying somewhat the capacity replacement makes it possible to "wait" for technologies and possibly reduce the overall production cost. In such a case, effective capacity credits should be taken into account according to the nature of the renewable resource and the level of penetration into the power mix. As a matter of fact, to replace a given capacity of dispatchable power, higher amounts of non-dispatchable resources are needed due to intermittency issues. Renewable capacity credits tend to decrease with higher penetration levels [START_REF] Keppler | Nuclear energy and renewables: system effects in low-carbon electricity systems[END_REF] .

V. CONCLUSIONS

In the French current electricity mix which is characterized by a high share of nuclear power, and in the context of current design of energy transition policies, the nuclear fleet replacement is a central issue.

To examine it with an academic viewpoint, we designed and proposed prospective scenarios in terms of renewable and nuclear power, together with nuclear fleet replacement scenarios.

Results showed that, according to the retained assumptions of the present study, implementing a progressive shut-down (i.e. with a longer lifetime of PWRs) appears a relevant choice since it both provides a lowest nuclear power production cost even at reduced load factors to participate in load following, and allows "waiting" for choosing most sustainable technologies.

In other words, even if the political guidelines are in favour of a reduction of electricity produced by nuclear plants, if nuclear power capacities are maintained so that it opens room for participation in load following, there is a strong interest in opting for a progressive replacement of PWRs by EPRs™ in order to cope with the intermittence of variable renewable resources. First studies demonstrated that, to reach a low-carbon power mix, both nuclear and renewables are needed, in a strongly-interconnected energy system perspective [START_REF] Caumon | Flexible hydrogen production implementation in the French power system: expected impacts at the French and European levels[END_REF] . In such a synergistic power system, both nuclear power plant flexibility and nuclear plant operation time extension should be examined, also by considering novel uses. New economic tools have to be built to address these important questions.

  Proceedings of Global 2015September 20-24, 2015 -Paris (France)Paper 5044

Fig. 1 .

 1 Fig. 1. PWR & EPR™ capacity evolution scenarios (for 50% nuclear penetration), own calculus

Fig. 2 .

 2 Fig. 2. Ratio of the nuclear annual levelized cost of electricity between the "prog." scenario and the "40 yrs" scenario (50% nuclear penetration), € 2010 /MWh / € 2010 /MWh, own calculus

Fig. 3 .

 3 Fig. 3. Residual load duration curves according to the renewable penetration, own calculus

Fig. 4 .

 4 Fig. 4. Example of assessment of the nuclear power modulation capacity ("high nuc" 2030), own calculus

  Proceedings of Global 2015September 20-24, 2015 -Paris (France)Paper 5044

  Proceedings of Global 2015September 20-24, 2015 -Paris (France)Paper 5044

TABLE III "

 III

	Baseload" nuclear contribution to satisfy the total power
	production according to the renewable penetration
	(% of produced energy), own calculus
	Renewable	Baseload Nuclear
	penetration (%)	penetration (%)
	16.5	57
	30	41
	50	17

TABLE IV Nuclear

 IV average load factor reduction for the total fleet, own calculus

	Scenario	2030	2050
	med	Ref	-9%
	high nuc	-5%	-19%
	high ren	-9%	-32%

ACKNOWLEDGMENTS

This work was carried out in the framework of a PhD thesis. The authors want to thank J. C. Bocquet, Head of the Laboratoire de Génie Industriel, and J. G. Devezeaux de Lavergne, Head of the Institute for techno-economics of energy systems (I-tésé) for the fruitful discussions.


Camille Cany
email: camille.cany@cea.fr

Christine Mansilla
email: christine.mansilla@cea.fr

Pascal Da Costa
email: pascal.da-costa@ecp.fr

Gilles Mathonnière
email: gilles.mathonnière@cea.fr

Jean-Baptiste Baptiste Thomas
email: jean-baptiste.thomas@cea.fr

Ecp Lgi

Nuclear power: a promising back-up option to promote renewable penetration in the French power system? Topic: Renewable system integration

Keywords: Renewable system integration renewable energy penetration, non-dispatchable renewable sources, back-up, low-carbon sources synergy

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The 3X20 European directive proposes renewable penetration goals [1]. In France, 27% of the electricity is to be produced by renewable resources (RES) by 2020 [START_REF]de l'Energie, du Développement Durable et de la Mer[END_REF] and this share will be continuously growing up to 2050. Among these resources, some -like wind and solar-are not dispatchable, which triggers challenges to maintain the reliability target level of the power system, both in the short and long term [START_REF] Gross | The costs and impacts of intermittency: an assessment of the evidence on the costs and impacts of intermittent generation on the British electricity network[END_REF], [START_REF] Hart | The potential of intermittent renewables to meet electric power demand: current methods and emerging analytical techniques[END_REF]. Non-dispatchable renewable resources currently benefit from a dispatch priority. The increase of their share in the power system modifies the residual demand (which is equal to the demand minus non-dispatchable renewable production) pattern to more power ramping and higher variation amplitudes. To keep the system balanced, back-up options are numerous, though scarcely considered exhaustively in power systems modelling (e.g. [START_REF] Davis | Determining the impact of wind on system costs via the temporal patterns of load and wind generation[END_REF], [START_REF] Keppler | Nuclear energy and renewables: system effects in low-carbon electricity systems[END_REF]). Besides peaking unit production -such as gas turbines-which is usually the preferred option, storage, demand (or supply) curtailment, interconnections and baseload (nuclear for instance) power modulation should also be considered, particularly in such a context of high non-dispatchable renewable penetration [START_REF] Hand | .1: Exploration of high-penetration renewable electricity futures[END_REF]. Considering the French power system, in order to maintain its reliability target level, nuclear power could participate to system balancing thanks to characteristics compatible with power modulation [START_REF] Bruynooghe | Load following operating mode at nuclear power plants and incidence on operation and maintenance costs. Compatibility with wind power variability[END_REF]- [START_REF] Grard | The nuclear fleet: a decisive enabler to integrate a share of renewables (in the framework of a comprehensive systems solution)[END_REF]. Increasing non-dispatchable renewable penetration levels would entail higher balancing needs [START_REF] Wagner | Feature of an electricity supply system based on variable input[END_REF], hence lesser outputs and operational modifications for nuclear power (e.g. higher fuel use, more maintenance needs, increased material and components ageing) [START_REF] Lokhov | Technical and economic aspects of load following with nuclear power plants[END_REF]. This paper investigates the overall economic impact of such an operating mode and compares nuclear and gas power load following options up to 2050, based on production costs and CO2 emission criteria. Several non-dispatchable renewable penetration level scenarios have been developed for France that take into account both wind and solar whereas most studies only focus on wind and hardly ever consider nuclear (e.g. [START_REF] Gross | The costs and impacts of intermittency: an assessment of the evidence on the costs and impacts of intermittent generation on the British electricity network[END_REF], [START_REF] Davis | Determining the impact of wind on system costs via the temporal patterns of load and wind generation[END_REF]). These scenarios are first detailed (section II). Then the methodology is presented (section III), followed by the results and discussion (section IV).

Power system scenarios

In 2013, the French Government launched a broad national consultation on the energy transition [START_REF]Ministère de l'Ecologie, du Développement Durable et de l'Energie[END_REF]. In this framework, several institutions designed energy prospective scenarios for France. The Institute for techno-economics of energy systems of the Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA/I-tésé) was involved in the contribution of the ANCRE (Alliance Nationale de Coordination de la Recherche pour l'Energie, the French National Alliance for Energy Research Coordination), that proposed three contrasted scenarios that all satisfy the factor 4, i.e. a reduction by four of the greenhouse gas emissions in 2050, compared to the 1990 level [START_REF] Ancre | Scénarios de l'ANCRE pour la transition énergétique[END_REF]. The specificity of these scenarios is a voluntaristic approach in terms of scientific and technological innovation. All these scenarios propose a similar non-dispatchable renewable penetration for a given time horizon, leading to a maximum energy penetration of 30% by 2050 (wind and solar). As regards nuclear power, two cases are considered: a 50% value (of the annual electricity production), in line with the French Government policy, and a "high nuclear" case with a 65% value. In order to consider a more voluntaristic scenario in terms of non-dispatchable renewable penetration, we also considered the scenario proposed by the ADEME (Agence de l'Environnement et de la Maîtrise de l'Energie, the French Environment and Energy Management Agency), which examines the case of 50% non-dispatchable renewable penetration by 2050 [START_REF] Ademe | L'exercice de prospective de l'ADEME : Vision 2030-2050[END_REF]. The ADEME also considered lower nuclear penetration (down to 20%). Furthermore, as regards nuclear power contribution to the French electricity mix, we also considered a "business-as-usual" value of 75%, which corresponds to the nuclear current contribution to the French annual electricity production [START_REF] Rte | Bilan électrique 2013[END_REF]. Finally, we established generic scenarios by crossing the renewable and nuclear penetration assumptions (cf. Table I and Table II), and studied each of them. For readability issues, only three contrasted cases will be presented in the results and discussion section. They provide conceivable target capacities in line with the current energy policy as regards RES development and the specific French context characterized by the significant place of nuclear power in the power mix:

1. "median scenario" (med): 16.5% wind and solar in 2030, 30% in 2050, 50% nuclear in both cases; 2. "high nuclear scenario" (high nuc): 65% nuclear, wind and solar shares as in the median case; 3. "high non-dispatchable renewable scenario" (high ren): 30% wind and solar in 2030, 50% in 2050, 50% nuclear in both cases.

Nuclear penetration

Reference scenario 75 % Business-as-usual 65 % High nuclear in ANCRE [START_REF] Ancre | Scénarios de l'ANCRE pour la transition énergétique[END_REF] 50 % Low nuclear in ANCRE [START_REF] Ancre | Scénarios de l'ANCRE pour la transition énergétique[END_REF] or High nuclear in ADEME [START_REF] Ademe | L'exercice de prospective de l'ADEME : Vision 2030-2050[END_REF] 20 % Low nuclear in ADEME [START_REF] Ademe | L'exercice de prospective de l'ADEME : Vision 2030-2050[END_REF] In order to assess the economic impact of nuclear power modulation operation, two scenarios have been defined as regards the French operational nuclear fleet (Pressurized Water Reactors (PWR) and Evolutionary Power Reactors (EPR ™ ) capacity evolution is shown on Fig. 1):

• Current reactors are operated 40 years and then shut down ("40 yrs" scenario);

•

The current fleet is progressively replaced to avoid massive investment in a short period of time: 2 GWe are shut down each year ("prog."; it is the "best-estimate" scenario). In each case, the current fleet is completed by EPR ™ in order to achieve the target nuclear penetration level of the selected scenario (cf. Fig. 1).

Figure 1. PWR & EPR

™ capacity evolution scenarios (for 50% nuclear penetration), own calculus

These scenarios have been established based on realistic expectations (the operating life extension of nuclear power plants is currently discussed in France) [START_REF]Nucléaire: EDF détaille les coûts de son «grand carénage[END_REF] and will result in different production costs according to the composition of the nuclear mix (the Capex of the nuclear mix is computed from the share between amortized power plants (PWR) and recently installed ones (EPR ™). Opex and life extension costs have also been included. They are issued from the French Court of Auditors in-depth study of the nuclear costs in the French context [START_REF]Les coûts de la filière électronucléaire[END_REF]).

Methodology: an approach based on the construction of annual residual load duration curves

To assess the impacts of non-dispatchable renewable resources on other power plants capacity requirements and annual productions in a power system, two major types of approaches can be distinguished. First, the construction of load net of wind and solar (called "residual" in what follows) duration curves for different wind and solar penetration levels allows determining the optimal mix of other generation resources in the long term [START_REF] Davis | Determining the impact of wind on system costs via the temporal patterns of load and wind generation[END_REF], [START_REF] Keppler | Nuclear energy and renewables: system effects in low-carbon electricity systems[END_REF], [START_REF] Ueckerdt | System LCOE: What are the costs of variable renewables?[END_REF]. Secondly, a minimum cost economic dispatch model can serve refining the general trends obtained from the first approach [START_REF] Davis | Determining the impact of wind on system costs via the temporal patterns of load and wind generation[END_REF], [START_REF] Keppler | Nuclear energy and renewables: system effects in low-carbon electricity systems[END_REF], [START_REF] Connolly | Modelling the existing Irish energy-system to identify future energy costs and the maximum wind penetration feasible[END_REF], [START_REF] Barnacle | Modelling generation and infrastructure requirements for transition pathways[END_REF]. The use of such a time step model allows short term supply and demand balancing in response to technical, economic and political constraints. We focus here on the first approach to put a light on general trends in the long term.

The developed methodology proceeds in several steps. On the one hand, the construction of annual residual load duration curves allows assessing the annual energy productions and optimal installed capacities, among which the needs for baseload installed power that will be fulfilled by nuclear in the French case. On the other hand, nuclear cycling capacity is evaluated based on the realistic French nuclear fleet scenarios that were presented in the previous section. Thanks to the annual residual load duration curves, we evaluate and compare nuclear fleet effective full power hours of both options: nuclear power fleet operated as a baseload capacity or as a combined option: baseload and cycling capacity. We then carry out the economic comparison. Finally, through the same approach, the nuclear load following option is compared to the gas one, which is considered as the reference back-up option.

Construction of annual residual load duration curves

A load duration curve represents the sorted hourly load of one year, starting with the highest load hour. The residual load curve is obtained by withdrawing the wind and solar production to the total annual production (considered constant on an annual cumulated basis in this study, hourly variations have been computed into the residual load duration curve based on the historical data from the French Transmission System Operator (RTE)). Wind and solar production was estimated by calculating mean hourly power values of historical data (between 2010 and 2013) also from RTE [START_REF]RTE database[END_REF], and by calibrating their hourly power to fit their assumed energy penetration level to satisfy a given demand (cf. Table I). The resulting curves are displayed on Fig. 2.

Assessment of the "optimal" nuclear power generation and the nuclear cycling capacity

The construction of screening curves (cf. Fig. 3) allows estimating the lowest total cost technology for a given operating time. Thus, they provide the optimal annual operating time for each resource type (e.g. 7900 hours for nuclear on the example given by Fig. 3), which can then be copied on the residual load duration curve, to assess the corresponding optimal installed power.

Baseload installed power will be fulfilled by nuclear in the French case. Other unavoidable energy productions capacities have been taken into account nonetheless (e.g. run-of-river hydroelectric plant).

Figure 3. Screening curve example (cost data from [START_REF] Salvadores | Projected costs of generating electricity[END_REF], 8% discount rate, no carbon price), own calculus While screening curves are commonly used to establish the optimal electric mix, our analysis showed that the results are extremely sensitive to the assumptions in terms of investment costs (are the interests during construction taken into account? What is the discount rate?), and CO2 price, only to name a few. When considering nuclear investment (very capex-intensive and low-carbon), these assumptions are critical. According to our calculations, when varying the discount rate of 3 points, or when taking into account a carbon price of $30/tonCO2, the optimal operating time of nuclear power could vary between 3000 and 8000 hours, leading to a resulting installed capacity between 55 GW and 40 GW, for 16.5% renewable penetration for instance.

Given this uncertainty range, we retained an empirical approach based on the historical availability factor of nuclear in France (return on operating experience) [START_REF]Power Reactor Information System (PRIS) database[END_REF]. The selected value is 7000 hours. Such an approach rather minimizes the resulting installed nuclear capacity. The optimal nuclear power will be derived by confronting this economical operating duration to the residual load curve that was constructed as explained in the previous section. By comparing the optimal nuclear power that corresponds to a given total annual production and non-dispatchable renewable energy penetration, to the installed power provided by the selected scenario, the available cycling power is assessed (cf. Fig. 4).

Assessment of the economic and environmental impacts

The economic impact can be assessed from two viewpoints:

• From the power plant operator viewpoint: the power plant does not supply all the available energy (cf. Fig. 4) and cycling operation induces overcosts (to be included in future studies). This will be assessed by comparing the levelized cost of electricity (LCOE, [START_REF] Salvadores | Projected costs of generating electricity[END_REF]) of the installed nuclear power either operated as a baseload or as a cycling capacity as imposed by the residual load (as shown in Fig. 4);

•

From the social welfare viewpoint: if not provided by nuclear, load following would have been carried out by other means. In this study we chose to compare to the gas power plant reference. The comparison is done by confronting the total cost of supplying the "cycling power" part (in blue in Fig. 4) either by nuclear or by gas. In the latter case, CO2 emissions will also be assessed and internalized in the electricity cost.

Cost assumptions as regards nuclear plants are issued from the French Court of Auditors [START_REF]Les coûts de la filière électronucléaire[END_REF] while the OECD assumptions have been adapted to the French case for the gas-fired power plants [START_REF] Salvadores | Projected costs of generating electricity[END_REF]. The open cycle gas turbine (OCGT) was selected as the reference due to its cycling abilities.

Results and discussion: The impact of renewable penetration on nuclear power generation -cost and environmental issues

As explained in the previous section, the installed nuclear power was confronted to the "optimal" value for each scenario, in order to assess the cycling possibility. Firstly, scenarios are compared through the corresponding optimal installed nuclear power. Table III shows the share of the total electricity production that would be provided by nuclear (for an economically optimal mix), according to the renewable penetration. From this table, we can point out that, for 16.5% renewable penetration, 57% of the electricity should be supplied by nuclear in the optimal case. Hence, 50% nuclear penetration (corresponding to the "med" scenario in 2030), is sub-optimal: no nuclear cycling capacity is available. Concerning the nuclear production cost, the two scenarios of nuclear fleet composition are compared in Fig. 5 ("prog" and "40 yrs"). Implementing a progressive shut-down first enables making profit from amortized PWRs (cycling is even less expensive than baseload operation from 2030 to 2050). It remains less expensive than the "40 yrs" scenario, even once amortized PWRs have been shut down. Indeed EPRs ™ that are installed later are characterized by higher availability, which results in smaller installed capacity needs, hence a reduced investment. Figure 5. Nuclear production cost according to the nuclear fleet composition scenario ("40 yrs" or "prog"), own calculus Since the "prog" scenario corresponds to the more realistic one in the French context, we will focus on it. For a given nuclear penetration level, the share of nuclear energy which is devoted to modulation increases more than linearly with the renewable penetration level. It is worth noticing that for the "high ren" scenario in 2050, 50% of nuclear energy would be dedicated to modulation. Fig. 6 shows the impact of modulation (loss of load factor) on the nuclear LCOE. The impact of the nuclear fleet composition on nuclear LCOE is pointed out by comparing the "med" 2050 case to the "high ren" 2030 case: similar nuclear and renewable penetration levels end up in different relative extra production costs. Indeed, in 2050, nuclear fleets are only composed of EPR™ whatever the renewable penetration scenario, while in 2030, relative costs are strongly dependent on the PWR replacement rate according to nuclear energy penetration targets. Figure 6. Load modulation impact on the nuclear LCOE, own calculus Finally, results show that the gas cycling option is cheaper than nuclear back-up in each scenario. For a given CO2 price value, the gas price which leads to equalize the cost of both options is determined. Conversely, the same approach is followed for a given gas price value. For all the scenarios the results are in the same order of magnitude: i/ for a CO2 price of €5/tonCO2 (which corresponds to the current market value [START_REF]CO2 market website[END_REF]), nuclear back-up is competitive when the gas price exceeds €35/MWh; ii/ for a gas price of €31/MWh (which corresponds to the ANCRE assumption for 2030 [START_REF] Ancre | Scénarios de l'ANCRE pour la transition énergétique[END_REF]), the gas option dominates as long as the CO2 price is lower than €20/tonCO2; iii/ if the combined cycle power plant (CCGT) was considered as the gas option (less flexible, yet probably enough), the previous results would be modified to €50/MWh and €110/tonCO2 respectively.

Overall, the gas option, even when cheaper would generate CO2 emissions in the range of 10-50Mt per year, depending on the scenario and the time horizon.

Conclusion and perspectives

Nuclear modulation seems to be achievable, especially in the case of France since a large production variation could be reached only by small increments in each power plant. This would remain true as long as installed nuclear power is large enough, in terms of contribution to the global electricity mix. In economic terms, gas back-up is more competitive than nuclear in each of the studied scenarios. However a carbon-tax could smooth the gap. When compared to open cycle gas turbines, a value as low as €20/tonCO2 could be enough to change the trends, which advocates for such incentives to avoid the effective greenhouse gas release. Nuclear back-up is all the more competitive than amortized power plants are utilized. Further studies could focus on assessing the impact of an increasing electricity demand, or technological progress by also including the overcosts due to cycling operation (both for the nuclear and gas options). When increasing the nondispatchable energy penetration, other back-up options, such as storage, curtailment, or demand-side management tools, should be considered as contributors to the "back-up" mix whose optimum remains to be established. This could be done through the use of a minimum cost economic dispatch model. Finally, nuclear modulation was apprehended via the concept of load following. Another way to reach the same target could be modulating through the output. In this case, the nuclear power plant would be operated as a baseload and electricity would be supplied to the electric system when requested. For the remaining time, instead of reducing the plant load, the output could be used to produce other valuable products such as heat or hydrogen [START_REF] Ruth | Nuclear-renewable hybrid energy systems: opportunities, interconnections and needs[END_REF]. The competitiveness of these productions is still to be proven and will be the topic of future works.

Figure 2 .

 2 Figure 2. Residual load duration curves, own calculus

Figure 4 .

 4 Figure 4. Example of assessment of the nuclear power modulation capacity ("high nuc" 2030), own calculus

TABLE I .

 I RENEWABLE ENERGY PENETRATION SCENARIOS

	Time horizon	Wind penetration	Solar penetration	Total (%)	Reference scenario
	2015	5 %	1 %	6 %	ANCRE [14]
	2020	8 %	2 %	10 %	ANCRE [14]
	2025	10 %	3 %	13 %	ANCRE [14]
	2030	12 %	4.5 %	16.5 %	ANCRE [14]
	2040	16 %	7 %	23 %	ANCRE [14]
	2050	20 %	10 %	30 %	ANCRE [14]
	2050	30 %	15 %	45 %	ADEME [15]
	2050	35 %	15 %	50 %	ADEME [15]
		TABLE II.	NUCLEAR ENERGY SCENARIOS

TABLE III .

 III NUCLEAR CONTRIBUTION TO SATISFY THE TOTAL ELECTRICITY PRODUCTION IF THE OPTIMAL MIX WAS IMPLEMENTED,

	ACCORDING TO THE RENEWABLE PENETRATION
	Renewable penetration (%)	Nuclear penetration (%)
	16.5	57
	30	41
	50	17

Acknowledgment

This work was carried out in the framework of a PhD thesis. The authors want to thank J. C. Bocquet, Head of the Laboratoire de Génie Industriel, and J. G. Devezeaux, Head of the Institute for techno-economics of energy systems (I-tésé) for the fruitful discussions.


Frédéric Bayart

Andreas Defant

Sunke Schlüters

Monomial convergence for holomorphic functions on ℓ r

published or not. The documents may come

Monomial convergence for holomorphic functions on ℓ_r

Introduction

Let X be a Banach sequence space (i.e., ℓ 1 ⊂ X ⊂ c 0 such that the canonical sequences (e k ) form a 1-unconditional basis) and R ⊂ X a Reinhardt domain (i.e., a nonempty open set such that any complex sequence u belongs to R whenever there exists z ∈ R with |u| ≤ |z|; for instance, the open unit ball B X of X ). Then each holomorphic (i.e., Fréchet differentiable) function f : R → C has a power series expansion α∈N n 0 c (n) α z α on every finite dimensional section R n of R, and for example from the Cauchy formula we can see that c (n) α = c (n+1)

α for α ∈ N n 0 ⊂ N n+1 0 .
Thus there is a unique family (c α ( f )) α∈N (N) 0 such that, for all

n ∈ N and all z ∈ R n , f (z) = α∈N (N) 0 c α z α .
The power series α c α z α is called the monomial expansion of f , and c α = c α ( f ) are its monomial coefficients.

Contrary to what happens on finite dimensional domains, the monomial expansion of f does not necessarily converge at every point of R. This in [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF] motivated the introduction of the following definition: Given a subset F (R) of H(R), the set of all holomorphic functions on R, we call mon F (R) = z ∈ R :

α∈N (N) 0 c α ( f )z α < ∞ for all f ∈ F (R)
the domain of monomial convergence with respect to F (R).

By continuity of a holomorphic function, and since the equality is satisfied on R n , we know that for all z ∈ mon F (R),

f (z) = α∈N (N) 0 c α ( f )z α .
We are mostly interested in determining mon F (R) when F (R) = P ( m ℓ r ) or H ∞ (B ℓ r ) for 1 ≤ r ≤ ∞; as usual we denote by H ∞ (B X ) the Banach space of all bounded holomorphic functions f : B X → C, and by P ( m X ) its closed subspace of all m-homogeneous polynomials P (i.e., all restrictions of bounded m-linear forms on X m to their diagonals).

The case r = 1 was solved completely by Lempert in [START_REF] Lempert | The Dolbeault complex in infinite dimensions II[END_REF], and the case r = ∞ seems fairly well-understood through the results of [START_REF] Bayart | Monomial series expansion of H p -functions and multipliers of H p -Dirichlet series[END_REF] (for more on these results see the introductions of the sections 5.1 and 5.2). However, for 1 < r < ∞, despite the results of [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF], the description of mon P ( m ℓ r ) and mon H ∞ (B ℓ r ) remains mysterious. In this paper, we improve the knowledge on these cases.

Of course, for X = ℓ 1 the fact that each sequence in ℓ 1 by definition is absolutely summable is a big advantage, and for X = ℓ ∞ the crucial tool is the Bohnenblust-Hille inequality (an inequality for m-linear forms on ℓ ∞ ) together with all its recent improvments. But for X = ℓ r with r = 1, ∞ we need alternative techniques.

The problem is to find for each u ∈ B ℓ r an additional summability condition which guaranties full control of all sums α c α ( f )u α , f ∈ H ∞ (B ℓ r ) . The general idea is simple. Split the set N (N) 0 of all multi indices α into a union of finite sets Λ n , and then each Λ n into the disjoint union of all its m-homogeneous parts Λ n,m (i.e., all α ∈ Λ n with order |α| = m). The challenge now is as follows: Find a clever decomposition

N (N) 0 = m,n Λ m,n , (1)
which allows a in a sense uniform control over all possible partial sums

α∈Λ n,m c α ( f )u α , f ∈ H ∞ (B ℓ r ) , (2)
such that under the additional summability property of u ∈ B ℓ r we for all functions f finally can conclude that

α∈N (N) 0 c α ( f )u α ≤ n m α∈Λ n,m c α ( f )u α < ∞ .
In order to study domains mon F (R) of monomial convergence, the decomposition in (1) which for our purposes is crucial, is inspired by the work of Konyagin and Queffélec from [START_REF] Konyagin | The translation 1 2 in the theory of Dirichlet series[END_REF]

Preliminaries

We use standard notation from Banach space theory. As usual, we denote the conjugate exponent of 1 ≤ r ≤ ∞ by r ′ , i.e.

M (m, n) = j = ( j 1 , . . . , j m ) ; 1 ≤ j 1 , . . . , j m ≤ n = {1, . . . , n} m M (m) = N m M = N N and J (m, n) = j ∈ M (m, n) ; 1 ≤ j 1 ≤ • • • ≤ j m ≤ n J (m) = n J (m, n) J = m J (m).
For indices i, j ∈ M we denote by (i, j) = (i 1 , i 2 , . . . , j 1 , j 2 , . . . ) the concatenation of i and j. An equivalence relation is defined in M (m) as follows: i ∼ j if there is a permutation σ such that i σ(k) = j k for all k. We write |i| for the cardinality of the equivalence class [i]. Moreover, we note that for each i ∈ M (m) there is a unique j ∈ J (m) such that i ∼ j.

Let us compare our index notation with the multi index notation usually used in the context of polynomials. There is a one-to-one relation between J (m) and

Λ(m) = α ∈ N (N) 0 ; |α| = ∞ i =1 α i = m ;
indeed, given j, one can define α by doing α r = |{q | j q = r }|; conversely, for each α, we consider j α = (1, α 1 . . ., 1, 2, α 2 . . ., 2, . . . , n, α n . . ., n, . . . ). In the same way we may identify

Λ(m, n) = α ∈ N n 0 ; |α| = m with J (m, n). Note that |j α | = m! α! for every α ∈ Λ(m).
Taking this correspondence into account, for every Banach sequence space X the monomial series expansion of a m-homogeneous polynomial P ∈ P ( m X ) can be expressed in different ways (we write c α = c α (P ))

α∈Λ(m) c α z α = j∈J (m) c j z j = 1≤j 1 ≤...≤j m c j 1 ... j m z j 1 • • • z j m .
Given a Banach sequence space X and some index subset J ⊂ J , we write P ( J X ) for the closed subspace of all holomorphic functions f ∈ H ∞ (B X ) for which c j ( f ) = 0 for all j ∈ J \ J . Clearly, P ( m X ) = P ( J (m) X ). If J ⊂ J is finite, then P ( J X ) = span z j : j ∈ J , where z j for j = ( j 1 , . . . , j ℓ ) stands for the monomial z j : u → u j := u j 1 •. . . •u j ℓ . For J ⊂ J (m), we call

J * = j ∈ J (m -1); ∃k ≥ 1, (j, k) ∈ J
the reduced set of J .

Unconditionality

Given a compact group G, the Sidon constant of a finite set C of characters γ (in the dual group) is the best constant c ≥ 0, denoted by S(C ), such that for every choice of scalars c γ , γ ∈ C , we have that

γ∈C |c γ | ≤ c γ∈C c γ γ ∞ .
An immediate consequence of the Cauchy-Schwarz inequality is that

1 ≤ S(C ) ≤ |C | 1 2 .
For the circle groups G = T, T n and T ∞ different values are possible:

• A well-known result of Rudin shows that for the set C = {1, z, . . . , z n-1 } of characters on G = T we have, up to constants independent of n,

S(C ) ≍ n . (3)
• In [START_REF] Defant | The Bohnenblust-Hille inequality is hypercontractive[END_REF] it was proved that for every m, n the Sidon constant of the monomials C = {z α : α ∈ Λ(m, n)} on G = T n , up to the mth power C m of some absolute constant C , satisfies

S(C ) ≍ |Λ(m -1, n)| 1 2 . ( 4
)
• In contrast, a reformulation of a result of Aron and Globevnik [START_REF] Aron | Analytic functions on c 0 Revista Matematica[END_REF]Thm 1.3] shows that for every m the Sidon constant of the sparse set

C = {z m j : j ∈ N} fulfills S(C ) = 1 . (5)
Let us transfer some of these results into terms of unconditional bases constants of spaces polynomials on sequence spaces. Recall that a Schauder basis (x n ) of a Banach space X is said to be unconditional whenever there is a con-

stant c ≥ 0 such that k ε k α k x k ≤ c k α k x k for every x = k a k x k ∈ X and all choices of (ε k ) k ⊂ C with |ε k | = 1.
In this case, the best constant c is denoted by χ (x n ) and called the unconditional basis constant of (x n ). If such a constant doesn't exist, i.e. if the basis is not unconditional, we set χ (x n ) = ∞.

Given a Banach sequence space X and an index set J ⊂ J , such that the set C = {z j : j ∈ J } of all monomials associated with J (counted in a suitable way) forms an basis of P ( J X ), we write

χ mon P ( J X ) = χ(C ) .
If we interpret each of these monomials z j as a character on the group T ∞ , then a straightforward calculation (using the distinguished maximum modulus principle) proves that

S(C ) = χ mon P ( J ℓ ∞ ) .
A simple but useful lemma shows that χ mon P ( J ℓ ∞ ) is an upper bound of all χ mon P ( J X ) . Lemma 3.1. Let X be a Banach sequence space and let J ⊂ J , such that the monomials form a basis of P ( J X ). Then χ mon P ( J X ) ≤ χ mon P ( J ℓ ∞ ) .

Proof. Assume χ mon P ( J ℓ ∞ ) < ∞ (otherwise there is nothing to show). For P ∈ P ( J X ) and a fixed u ∈ B X define Q(w) = P (wu

) ∈ P ( J ℓ ∞ ). Since B X is a Reinhardt domain, we have Q ∞ ≤ P ∞ . It is now sufficient to observe that j∈J |c j (P )u j | = sup w ∈B ℓ ∞ j∈J |c j (P )u j ||w j | = sup w ∈B ℓ ∞ j∈J |c j (Q)||w j | ≤ χ mon P ( J ℓ ∞ ) Q ∞ ≤ χ mon P ( J ℓ ∞ ) P ∞ , the conclusion.
Let us again see some examples: Given X , an immediate consequence of ( 3) is that for P ( J X = span{z j 1 ; 0 ≤ j ≤ n -1} we have, up to a universal constant,

χ mon P ( J X ) ≍ n,
and from ( 5) we may deduce that for

J = (k, • • • , k) ; k ∈ N ⊂ J (m) χ mon P ( J X ) = 1.
Generalizing ( 4) is much more complicated. In the scale of all ℓ r -spaces the results from [START_REF] Bayart | Maximum modulus of random polynomials[END_REF] (lower estimates) and [START_REF] Defant | The Bohr radius of the unit ball of ℓ n p[END_REF][START_REF] Defant | The Bohnenblust-Hille inequality is hypercontractive[END_REF]

(upper estimates) show that for 1 ≤ r ≤ ∞ χ mon P ( J (m,n) ℓ r ) ≍ J (m -1, n) 1-1 min (r,2) , (6)
where ≍ means that the left and the right side equal up to the m-th power C m of a constant only depending on r (and neither on m nor on n).

Replacing the index set J (m, n) by an arbitrary finite subset J ⊂ J (m, n) the following result is a strong improvement and our main tool within our later study of sets of monomial convergence. Theorem 3.2. Given 1 ≤ r ≤ ∞ and m ≥ 1, there is a constant C (m, r ) ≥ 1 such that for every n ≥ 1, every P ∈ P ( J (m,n) ℓ r ), every J ⊂ J (m, n), and every u ∈ ℓ r we have

j∈J c j (P ) |u j | ≤ C (m, r )|J * | 1-1 min(r,2) u m r P ∞ , (7)
where

C (m, r ) ≤ eme (m-1)/r if 1 ≤ r ≤ 2 em2 (m-1)/2 if 2 ≤ r ≤ ∞.
In particular, for every finite J ⊂ J (m)

χ mon P ( J ℓ r ) ≤ C (m, r )|J * | 1-1 min(r,2) . ( 8
)
The proof is given in the following two subsections; it is different for r ≤ 2 and for r ≥ 2. The case r = ∞ of ( 6) is given in [START_REF] Defant | The Bohnenblust-Hille inequality is hypercontractive[END_REF], and it uses the hypercontractive Bohnenblust-Hille inequality. The general case 1 ≤ r ≤ ∞ from [START_REF] Defant | The Bohr radius of the unit ball of ℓ n p[END_REF] needs sophisticated tools from local Banach space theory (as Gordon-Lewis and projection constants). These arguments in fact only work for the whole index set J (m, n), and they seem to fail in full generality for subsets J of J (m, n). We here provide a tricky, but quite elementary, argument which works for arbitrary J ; moreover, we point out that even for the special case J = J (m, n) we obtain better constants C (m, r ) for ( 8) than in [START_REF] Defant | The Bohr radius of the unit ball of ℓ n p[END_REF].

From [START_REF] Defant | Unconditionality in spaces of m-homogeneous polynomials[END_REF] we know that for each infinite dimensional Banach sequence space X , the Banach space P ( m X ) never has an unconditional basis. In particular, the unconditional basis constant χ mon P ( m X ) of all monomials (z j ) j∈J (m) is not finite. But let us note that in contrast to this there are X such that for each m sup

n χ mon P ( J (m,n) X ) < ∞
(this can be easily shown for X = ℓ 1 , but following [START_REF] Defant | Unconditionality in spaces of m-homogeneous polynomials[END_REF] there are even examples of this type different from ℓ 1 ).

The case r ≤ 2

We need several lemmas. The first one is a Cauchy estimate and can be found in [7, p. 323]. For the sake of completeness we include a streamlined argument.

|c α (P )| ≤ m m α α 1 r P ∞ .
In particular, for each j ∈ J (m, n) we have that

|c j (P )| ≤ e m r |j| 1 r P ∞ . Proof. Define u = m -1/r (α 1/r 1 , . . . , α 1/r n ) ∈ B ℓ n r .
Then by the Cauchy integral formula for each

P ∈ P m ℓ n r c α (P ) = 1 (2πi ) n |z n |=u n . . . |z 1 |=u 1 P (z) z α z 1 . . . z n d z .
Hence we obtain

|c α (P )| ≤ 1 |u α | P ∞ = m m α α 1 r P ∞ , the conclusion. For the second inequality note first that m m α α 1 r ≤ e m/r m! α! 1/r
, and recall that if we associate to j the multi index α, then m! α! = |j| .

Corollary 3.4. Consider the linear operator

Q ∈ L ℓ n r , P ( m-1 ℓ n r ) defined by Q(z, w) = j∈J (m-1,n) n k=1 b (j,k) z j w k ,
where z, w ∈ ℓ n r . Then for any j

∈ J (m -1, n) , n k=1 |b (j,k) | r ′ 1/r ′ ≤ e m-1 r |j| 1/r Q ∞ . Proof. Let us fix w ∈ B ℓ n r . Then Q(•, w) ∈ P ( m-1 ℓ n r )
. Thus, by the preceding lemma for any j

∈ J (m -1, n), n k=1 b (j,k) w k ≤ e m-1 r |j| 1/r sup z∈B ℓ n r |Q(z, w)| ≤ e m-1 r |j| 1/r Q ∞ .
We now take the supremum over all possible w ∈ B ℓ n r . Lemma 3.5.

Let P ∈ P ( m ℓ n r ). Then for any j ∈ J (m -1, n) n k=j m-1 |c (j,k) (P )| r ′ 1/r ′ ≤ me 1+ m-1 r |j| 1/r P ∞ .
Proof. Let A : ℓ n r × . . . × ℓ n r → C be the symmetric m-linear form associated to P , A(z (1) , . . . , z (m) ) = i∈M (m,n) a i (A)z (1)

i 1 • • • z (m) i m ;
in particular, for each j ∈ J (m, n) we have a j (A) = c j (P) |j| . For z, w ∈ ℓ n r define the linear operator

Q(z, w) = A(z, . . . , z, w) ∈ L ℓ n r , P ( m-1 ℓ n r ) ; then a simple calculation proves Q(z, w) = i∈M (m,n) a i (A)z i 1 • • • z i m-1 w i m = i∈M (m-1,n) n k=1 a (i,k) (A)z i 1 • • • z i m-1 w k = j∈J (m-1,n) i∈[j] n k=1 a (i,k) (A)z i 1 • • • z i m-1 w k = j∈J (m-1,n) n k=1 i∈[j] a (i,k) (A)z i 1 • • • z i m-1 w k = j∈J (m-1,n) n k=1 a (j,k) (A)|j|z j 1 • • • z j m-1 w k .
Now note that for every j ∈ J (m -1, n) we have |(j, k)| ≤ m|j| , and hence by the preceding corollary for such j

k:j m-1 ≤k c (j,k) (P ) r ′ 1/r ′ = k:j m-1 ≤k a (j,k) (A)|(j, k)| r ′ 1/r ′ ≤ m n k=1 a (j,k) (A)|j| r ′ 1/r ′ ≤ m e m-1 r |j| 1/r Q ∞ .
Finally, by Harris' polarization formula we know that Q ∞ ≤ e P ∞ , and hence we obtain the desired conclusion.

Now we are ready to give the

Proof of Theorem 3.2 for 1 ≤ r ≤ 2. Take P ∈ P ( J (m,n) ℓ r ), J ⊂ J (m, n) and u ∈ ℓ r . Then, by Lemma 3.5, for any j ∈ J * ,

k: (j,k)∈J |c (j,k) (P )| r ′ 1/r ′ ≤ n k=j m-1 |c (j,k) (P )| r ′ 1/r ′ ≤ me 1+ m-1 r |j| 1/r P ∞ .
Now by Hölder's inequality (two times) and the multinomial formula we have

j∈J |c j (P )||u j | = j∈J * k: (j,k)∈J |c (j,k) ||u j ||u k | ≤ j∈J * |u j | k: (j,k)∈J |c (j,k) | r ′ 1/r ′ k |u k | r 1/r ≤ me 1+ m-1 r j∈J * |j| 1/r |u j | u r P ∞ ≤ me 1+ m-1 r j∈J * |j||u j | r 1/r j∈J * 1 1/r ′ u r P ∞ ≤ me 1+ m-1 r j∈J (m-1,n) |j||u j | r 1/r j∈J * 1 1/r ′ u r P ∞ = me 1+ m-1 r |J * | 1-1 r u m r P ∞ .
In order to deduce [START_REF] De La Bretèche | Sur l'ordre de grandeur des polynômes de Dirichlet[END_REF], note that for every finite J ⊂ J (m) there is n such that J ⊂ J (m, n). Then every P ∈ P ( J ℓ r ) can be considered as a polynomial in P ( J (m,n) ℓ r ) with equal norm, which implies the conclusion.

The case r ≥ 2

Note first that the simple argument from the proof of Lemma 3.1 shows that we only have to deal with the case r = ∞. For r = ∞ we need another lemma which substitutes the argument (by Cauchy's estimates) from Lemma 3.3. It is an improvement of Parseval's identity, and its proof can be found in [6, Lemma 2.5].

Lemma 3.6.

Let P ∈ P ( J (m,n) ℓ ∞ ). Then n k=1     j∈J (m-1,n) j m-1 ≤k |c (j,k) (P )| 2     1/2 ≤ em2 m-1 2 P ∞ .
We are now ready for the Proof of Theorem 3.2 for r = ∞. Let P ∈ P ( J (m,n) ℓ ∞ ). Then, for any u ∈ B ℓ ∞ , by the Cauchy-Schwarz inequality and the preceding Lemma 3.6 we have

j∈J |c j (P )||u j | ≤ n k=1     j∈J * (j,k)∈J |c (j,k) |     ≤ n k=1     j∈J * (j,k)∈J |c (j,k) | 2     1/2 {j ∈ J * : (j, k) ∈ J } 1/2 ≤ n k=1     j∈J (m-1,n) j m-1 ≤k |c (j,k) | 2     1/2 |J * | 1/2 ≤ em2 m-1 2 |J * | 1/2 P ∞ .
For the second statement, see again the argument from the proof in the case 1 ≤ r ≤ 2. This finally completes the proof of Theorem 3.2.

Remark 3.7. It is natural to ask for lower bounds of χ mon P ( J ℓ r ) using |J | or |J * |. For the whole set of m-homogeneous polynomials, this has been done in [START_REF] Defant | Bohr's power series theorem and local Banach space theory[END_REF] for r ≥ 2 and in [START_REF] Bayart | Maximum modulus of random polynomials[END_REF] for 1 ≤ r ≤ 2. Using the Kahane-Salem-Zygmund inequality, we can give such a lower bound at least for the case r = ∞. Indeed, assume that J ⊂ J (m, n). Then there exists some absolut constant C > 0 and signs (ε j ) j∈J such that sup

u∈B ℓ n ∞ j∈J ε j u j ≤ C n 1/2 |J | 1/2 (log m) 1/2 .
Now, the inequality

|J | = sup u∈B ℓ n ∞ j∈J |ε j ||u j | ≤ χ mon P ( J ℓ ∞ ) sup u∈B ℓ n ∞ j∈J ε j u j yields χ mon P ( J ℓ ∞ ) ≥ |J | 1/2 C n 1/2 (log m) 1/2 .
However, the inequality given by Theorem 3.2 is very bad if J involves many independent variables; see in particular [START_REF] Bayart | The Bohr radius of the n-dimensional polydisk is equivalent to (log n)/n[END_REF].

Remark 3.8. Given an index set J ⊂ J , we define the Bohr radius of a Reinhardt R in C n with respect to J by

K (R; J ) = sup 0 ≤ r ≤ 1 ∀ f ∈ H ∞ (R) : sup u∈r R j∈J c j ( f )u j ≤ f ∞ .
The standard multi-variable Bohr radius is then denoted by K (R) = K (R; J ). Let us recall the two most important results on Bohr radii: For the open unit disc R = D, Bohr's power series theorem states that K (D) = 1 3 , and in [START_REF] Bayart | The Bohr radius of the n-dimensional polydisk is equivalent to (log n)/n[END_REF](following the main idea of [START_REF] Defant | The Bohnenblust-Hille inequality is hypercontractive[END_REF]) it was recently proved that

lim n→∞ K (B ℓ n ∞ ) log n n = 1 .
For every 1 ≤ r ≤ ∞ and every n (with constants depending on r only) we have

K (B ℓ n r ) ≍ log n n 1-1 min{r,2} . (9)
The probabilistic argument for the upper estimate is due to [START_REF] Boas | Majorant series[END_REF] (see also [START_REF] Defant | Bohr's power series theorem and local Banach space theory[END_REF]), and the proof of the lower estimate from [START_REF] Defant | The Bohr radius of the unit ball of ℓ n p[END_REF] uses symmetric tensor products and local Banach space theory. We here sketch a simplified argument based on Theorem 3.2.

Theorem 3.9. Let 1 ≤ r ≤ ∞ and σ = 1-1 min(r,2) . Then there is a constant C = C (r ) such for every J ⊂ J and every n

C sup m (J (m, n) * σ m ≤ K (B ℓ n r ; J ) , ( 10
)
where J (m, n)

:= J ∩ J (m, n) and C ≥ 1 3e 2 > 0.
Proof. By a simple analysis of [10, Theorem 2.2] as well as [10, Lemma 2.1] we have

1 3 inf m K B ℓ n r ; J (m, n) ≤ K (B ℓ n r ; J ) , and
K B ℓ n r ; J (m, n) = 1 m χ mon P ( J (m,n) ℓ r )
.

Then the conclusion is an immediate consequence of Theorem 3.2 and the simple fact that for the constant C (m, r ) ≤ eme (m-1)/ min{r,2} ≤ e 2m . Now the proof for the lower bound in (9) follows from the the special case J = J . Indeed,

J * (m, n) = J (m -1, n) = (m -1) + n -1 m -1 ≤ e m-1 1 + n m -1 m-1
, hence inserting this estimate into [START_REF] Defant | Bohr's power series theorem and local Banach space theory[END_REF] and minimizing over m gives what we want.

The Konyagin-Queffélec method

We now apply Theorem 3.2 to a special method of summation which was originally used by Konyagin and Queffélec to find the correct asymptotic order of the Sidon constant of Dirichlet polynomials of lenght x. In [START_REF] Konyagin | The translation 1 2 in the theory of Dirichlet series[END_REF] they proved the following: there exists a constant β > 0, such that for every Dirichlet polynomial

x n=1 a n n -s , x n=1 |a n | ≤ x exp -β + o(1) log x log log x sup t∈R x n=1 a n n i t . ( 11
)
This was improved in [START_REF] De La Bretèche | Sur l'ordre de grandeur des polynômes de Dirichlet[END_REF], where an improved lower bound on β was given, and in [START_REF] Defant | The Bohnenblust-Hille inequality is hypercontractive[END_REF] where the precise value β = 1

It turns out that ( 11) is linked to our subject by the Bohr point of view. Indeed, define for each x the index set J (x) := j ∈ J : p j ≤ x . Then to each Dirichlet polynomial

D(s) = x n=1 a n n -s = j∈J (x) a p j p -s j ,
we can associate a polynomial

P (z) = j∈J (x) a p j z j ∈ P ( J ℓ ∞ ) .
Kronecker's theorem ensures that P ∞ = sup t∈R |D(i t )|, and the result of [14, Theorem 3] translates into the following remarkable equality (the improvement of (11) mentioned above):

χ mon P ( J (x) ℓ ∞ ) = x exp - 1 + o(1) log x log log x ; ( 12
)
in other terms, the latter expression gives the precise asymptotic order of the Sidon constant S(x) for the characters z j , j ∈ J (x) on the group T ∞ . There is also an m-homogeneous version of ( 12) due to Balasubramanian, Calado and Queffeléc [START_REF] Balasubramanian | The Bohr inequality for ordinary Dirichlet series[END_REF] with an original formulation analog to [START_REF] Defant | A logarithmic lower bound for multi-dimensional Bohr radii[END_REF]. We reformulate it as follows: Define for m the index set J (x, m) := j ∈ J (m) : p j ≤ x . Then with constants only depending on m

χ mon P ( J (x,m) ℓ ∞ ) ≍ x m-1 2m log x m-1 2 . ( 13
)
The following two theorems extend these results to the scale of ℓ r -spaces, and more. The original proofs of ( 12) and ( 13) are heavily based on the Bohnenblust-Hille inequality and its recent improvements. Here we need Theorem 3.2 as a substitute. Part (1) of the first theorem obviously extends the upper estimate from [START_REF] Defant | Bohr's strip for vector valued Dirichlet series[END_REF] to the scale of ℓ r -spaces, part (2) even modifies the index set J (x, m) (so far defined via primes). Both results are of particular interest for our study of sets of monomial convergence in the next section. . Then for every f ∈ H ∞ (B ℓ r ), every u ∈ B ℓ r , and every x > e we have [START_REF] Balasubramanian | The Bohr inequality for ordinary Dirichlet series[END_REF] for p denoting the sequence of primes,

j:p j ≤x c j ( f )u j ≤ x σ exp -2σ + o(1) log x log log x f ∞ .

13

(2) for q = (q k ) k , defined by q k = k • log(k + 2) θ with some θ ∈ ( 1 2 , 1],

j:q j ≤x c j ( f )u j ≤ x σ exp -2σ θ - 1 2 + o(1) log x log log x f ∞ .
In both cases, the o-term depends neither on x nor on f .

The second theorem extends [START_REF] Defant | The Bohr radius of the unit ball of ℓ n p[END_REF] to the scale of ℓ r -spaces.

Theorem 4.2. Let 1 ≤ r ≤ ∞ and m ≥ 1. Then there exists C (m, r ) > 0 such that for all P ∈ P ( m ℓ r ), all x ≥ 3, and all u ∈ ℓ r we have [START_REF] Balasubramanian | The Bohr inequality for ordinary Dirichlet series[END_REF] in the case 1 ≤ r ≤ 2:

j:p j ≤x c j (P )u j ≤ C (m, r ) x m-1 m (1-1 r ) log log x (m-1)(1-1 r ) (log x) (1-1 r ) u m r P ∞ ,
(2) and in the case 2 ≤ r ≤ ∞:

j:p j ≤x c j (P )u j ≤ C (m, r ) x m-1 2m log x m-1 2 u m r P ∞ .
Clearly all these results have reformulations in terms of unconditional basis constants. For example, part (1) of Theorem 4.1 reads:

χ mon P ( J (x) ℓ r ) ≤ x σ exp -2σ + o(1) log x log log x .
The proofs will be given in (4.2); the next section prepares them.

Size of some index sets

Although we stated Theorem 4.1 for the sequence of primes p in part (1) and for a specific choice of q in part (2) we want to state our considerations below as generic as possible. Let hereinafter q = (q k ) k denote a strictly increasing sequence with q 1 > 1 and q k → ∞ for k → ∞. For technical reasons we have to introduce the index of length zero ϑ = ( ), for which q ϑ = 1 and (i, ϑ) = (ϑ, i) = i by convention. Let x > 2 and 2 < y < x. Choose l ∈ N, such that q l ≤ y < q l +1 . We define

J (x) := j ∈ J q j ≤ x ∪ {ϑ} J -(x; y) := j = ( j 1 , . . . , j k ) ∈ J (k) k ∈ N, q j ≤ x, j k ≤ l ∪ {ϑ}
and for m ∈ N, J (x, m) := j = ( j 1 , . . . , j m ) ∈ J (m) q j ≤ x J + (x, m; y) := j = ( j 1 , . . . , j m ) ∈ J (x, m) l < j 1 , respectively for m = 0, J + (x, 0; y) := {ϑ}.

From the general construction of these sets we can already say something about their size -we need five lemmas. (1)

|J -(x; y)| ≤ 1 + log x log q 1 l (2) |J (x, m)| = whenever m > log x log q 1 .
Proof. [START_REF] Balasubramanian | The Bohr inequality for ordinary Dirichlet series[END_REF] Using the correspondence between J (m) and Λ(m), J -(x; y) has the same cardinal number as

Γ -(x; y) := α ∈ N l 0 q α 1 1 • • • q α l l ≤ x . Now, for α ∈ Γ -(x; y) and 1 ≤ j ≤ l , q α j 1 ≤ q α 1 1 • • • q α l l ≤ x,
so that α j ≤ log x log q 1 for all j . (2) Note that for every j ∈ J + (x, m; y) we have q m 1 ≤ q j ≤ x which immediately gives the conclusion.

The next lemma relates the size of an index set with the size of its reduced set.

Lemma 4.4. For the reduced index sets,

J (x, m) * ⊂ J x m-1 m , m -1 and J + (x, m; y) * ⊂ J + x m-1 m , m -1; y .
Proof. Let j = ( j 1 , . . . , j m-1 ) ∈ J (x, m) * , respectively j ∈ J + (x, m; y) * . Then there exists k ≥ j m-1 such that (j, k) ∈ J (x, m), respectively (j, k) ∈ J + (x, m; y). Hence

q j • q k = q (j,k) ≤ x. Since q k ≥ q j m-1 , this implies either q k > x 1 m or q j 1 ≤ . . . ≤ q j m-1 ≤ q k ≤ x 1 m . In both cases, q j 1 • • • q j m-1 ≤ x m-1 m .
For specific choices of q we can say the following about the size of J + (x, m; y): Lemma 4.5. Let q = (q k ) k be defined by q k = k • log(k + 2)

θ for some θ ∈ (0, 1].

Then there exists a constant c > 0, such that for every x > y > 2 and every m ∈ N,

|J + (x, m; y)| ≤ x y -m exp y • g θ (x) + c
where g θ (x) = 1 1-θ (log x) 1-θ for θ < 1 and g θ (x) = log log x for θ = 1.

Proof. From the definition of the series q, we see immediately

q l +k -q l ≥ q k ( 14
)
for any k ∈ N. We have furthermore for c = q -1 1 + q -1 2 + q -1 3 ,

k≤x 1 q k ≤ 3<k≤x 1 k(log k) θ + c ≤ x 3 1 t (log t ) θ dt + c = log x log 3 1 s θ ds + c
and therefore by integration

k≤x 1 q k ≤ g θ (x) + c. ( 15
)
We introduce a completely multiplicative function,

|J + (x, m; y)| = j∈J + (x,m;y) 1 ≤ x y m j∈J + (x,m;y) y q j 1 • • • y q j m ≤ x y m l <k<x ∞ ν=1 y q k ν ≤ x y m exp - l <k<x log 1 - y q k .
Using the series expansion of the logarithm around 1, we obtain for the exponent

- l <k<x log 1 - y q k = l <k<x ∞ ν=1 1 ν y q k ν ≤ l <k<x y q k 1 1 - y q k .
With [START_REF] Defant | The Bohnenblust-Hille inequality is hypercontractive[END_REF] and the fact that y ≥ q l , this leads to

- l <k<x log 1 - y q k ≤ y l <k<x 1 q k -y ≤ y l <k<x 1 q k-l ≤ y • k<x 1 q k .
(15) now completes the proof.

Finally, we mention two known estimates which measure the size of J (x, m) and J + (x, m; y), in the case they are defined with respect to the sequence of primes. The first one is taken from Balazard [3, Corollaire 1], and the second one is a well-known result of Landau (see e.g. [START_REF] Hardy | An introduction to the theory of numbers[END_REF] for a proof).

Lemma 4.6. Let q denote the sequence of primes. Then there exists a constant c > 0, such that for every x > y > 2 and every m ∈ N,

|J + (x, m; y)| ≤ x y -m exp y • log log x + c .
Lemma 4.7. Let q denote the sequence of primes and let m ≥ 1. There exists a constant C m > 0 such that, for all x ≥ 3,

|J (x, m)| ≤ C m x log x (log log x) m-1 (16)

Proofs

The proof of Theorem 4.2 is now very short.

Proof of Theorem 4.2. The proof of the first statement is a direct consequence of Theorem 3.2 for the index set J = J (x, m), and of the Lemmas 4.4 and 4.7. The second statement follows from Lemma 3.1 combined with [START_REF] Defant | The Bohr radius of the unit ball of ℓ n p[END_REF].

To present the Konyagin-Queffélec technique in general we need one more additional lemma.

Lemma 4.8. Let m 1 , m 2 , l ∈ N and P ∈ P ( m 1 +m 2 ℓ r ) such that c k (P ) = 0 for only finitely many k ∈ J (m 1 + m 2 ). Then for every i ∈ J (m 1 , l ) the polynomial

P i = j∈J (m 2 ) j 1 >l c (i,j) (P ) z (i,j) ∈ P ( m 2 ℓ r ) satisfies P i ∞ ≤ P ∞ .
Proof. Given u ∈ ℓ r , a straightforward calculation shows

P i (u) = T l P ζ 1 u 1 , . . . , ζ l u l , u l +1 . . . ζi 1 • • • ζi l d(ζ 1 , . . . , ζ l ) ,
which immediately implies the desired inequality.

Proof of Theorem 4.1. Recall the setting of our theorem. Let x > e and 2 < y < x, and choose l ∈ N such that q l ≤ y < q l +1 . Given u ∈ B ℓ r , at first write u = u -+u + where u - k = 0 for k > l and u + k = 0 for k ≤ l . Any k ∈ J (x) may be written as k = (i, j) with i ∈ J -(x; y) and j ∈ J + (x, m; y). Moreover,

|u i | = |u - i | and |u j | = |u + j |. Hence, q k ≤x |c k u k | = i∈J -(x;y) m∈N 0 j∈J + (x,m;y) q (i,j) ≤x |c (i,j) u (i,j) | = i∈J -(x;y) m∈N 0 |u - i | j∈J + (x,m;y) q (i,j) ≤x |c (i,j) u + j | .
Using Theorem 3.2, we can now estimate the latter sum for every i ∈ J -(x; y),

|u - i | j∈J + (x,m;y) q (i,j) ≤x |c (i,j) u + j | ≤ |u - i |C m |J + (x, m; y) * | σ sup ζ r ≤ u + r ∀k≤l :ζ k =0 j∈J (m) j 1 >l c (i,j) ζ j ≤ C m |J + (x, m; y) * | σ sup ζ r ≤ u + r ∀k≤l :ζ k =0 j∈J (m) j 1 >l c (i,j) u - i ζ j ≤ C m |J + (x, m; y) * | σ j∈J (m) j 1 >l c (i,j) z (i,j) ∞ ,
where the last inequality is a consequence of (u

-+ ζ) (i,j) = u - i ζ j and u -+ ζ r r = u -r r + ζ r r ≤ u -r r + u + r r ≤ 1 .
Choose for each i ∈ J -(x, y) some m i ∈ N such that i ∈ J (m i ). By Lemma 4.8 we then obtain

q k ≤x |c k u k | ≤ i∈J -(x;y) m C m |J + (x, m; y) * | σ k∈J (m+m i ) c k z k ∞ .
Moreover, if we decompose f into its sum of homogeneous Taylor polynomials, then we deduce by Cauchy estimates that

C m |J + (x, m; y) * | σ ≤ 1 + log x log q 1 l +1 sup m C m |J + (x m-1 m , m -1)| σ .
Up to this point, our arguments are independent of the specific choice of q. We threat both cases at once. In the case of q denoting the sequence of primes, set θ = 1. By Lemma 4.5 and Lemma 4.6, respectively

1 + log x log q 1 l +1 • sup m C m |J + (x m-1 m , m -1)| σ ≤ 1 + log x log q 1 l +1 • sup m C m x m-1 m y -m+1 exp y • g θ (x) + c σ .
Choosing y =

Monomial convergence

In this section we apply the new estimates on the unconditional basis constant of polynomials on ℓ r from the preceding two sections, to the analysis of sets mon P ( m ℓ r ) and mon H ∞ (B ℓ r ) of monomial convergence of m-homogeneous polynomials on ℓ r and bounded holomorphic functions on B ℓ r .

Polynomials

The next statement gives the state of art for homogeneous polynomials. (1

) If r = ∞, then mon P ( m ℓ ∞ ) = ℓ 2m m-1 ,∞ .
In particular, for all ε > 0, 1 p σ m +ε • ℓ r ⊂ mon P ( m ℓ r ).

Proof. Let P = j∈J (m) c j (P )z j ∈ P ( m ℓ r ) and let u ∈ ℓ r . We intend to show that

S := j∈J (m) |c j (P )| 1 (p j 1 • • • p j m ) σ m log(p j 1 ) • • • log(p j m ) ε |u j | ≤ C u m r P ∞
for some constant C > 0. Let us observe that, for any j 1 , . . . , j m ≥ 1,

log(p j 1 ) • • • log(p j m ) ≥ (log 2) m-1 m log(p j 1 • • • p j m ). (17)
We order the sum over j ∈ J (m) with respect to the value of the product p j 1 • • • p j m . Precisely, using (17), we write

S ≪ +∞ N=m j∈J (m) 2 N ≤p j <2 N +1 1 p σ m j log ε (p j ) |c j (P )||u j | ≪ +∞ N=m 1 2 Nσ m N ε p j ≤2 N +1 |c j (P )||u j |.
We apply Theorem 4.2 to find

S ≪ +∞ N=m 1 2 Nσ m N ε 2 Nσ m log(N ) (m-1) 1-1 r N 1-1 r P ∞ u m r .
The series is convergent since ε > 1/r .

Finally, we are ready to provide the Proof of the lower inclusion of Theorem 5.1, (4). Given u ∈ ℓ (mr ′ ) ′ -ε , we show that the decreasing rearrangement u * ∈ mon P ( m ℓ r ). Then for some δ > 0 we have

u * n ≪ 1 n 1 (mr ′ ) ′ -ε = 1 n 1 (mr ′ ) ′ +δ
.

By the prime number theorem we know that p n ≍ n log n, hence Remark 5.4. A look at [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF] shows that in the case r > 2 the proof of the inclusion mon

1 n 1 (mr ′ ) ′ +δ = 1 p m-1 m 1 r ′ + δ 2 n p m-1 m 1 r ′ + δ 2 n n 1 (mr ′ ) ′ +δ ≪ 1 p m-1 m 1 r ′ + δ 2 n (n logn) m-1 m 1 r ′ + δ 2 n 1 (mr ′ ) ′ +δ . But obviously (n logn) m-1 m 1 r ′ + δ 2 n 1 (mr ′ ) ′ +δ = (log n) m-1 m 1 r ′ + δ 2 n 1 r n δ 2 ∈ ℓ r ,
P ( m ℓ r ) ⊂ ℓ m-1 2m + 1 r -1
,∞

keeps working if we replace ℓ r by ℓ r,∞ . Indeed, it just uses that sup

u∈ℓ n r , u r ≤1 n k=1 |u k | 2 = n 1-
( m ℓ r,∞ ) = ℓ m-1 2m + 1 r -1
,∞ .

Holomorphic functions

We now study mon H ∞ (B ℓ r ) for 1 ≤ r ≤ +∞. The extreme cases are already well-known: By a result of Lempert (see e.g. [START_REF] Lempert | The Dolbeault complex in infinite dimensions II[END_REF] and [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF]) we have

mon H ∞ (B ℓ 1 ) = B ℓ 1 . (18)
Moreover by [START_REF] Bayart | Monomial series expansion of H p -functions and multipliers of H p -Dirichlet series[END_REF] we know that

B ⊂ mon H ∞ (B ℓ ∞ ) ⊂ B ( 19
)
where

B = u ∈ B ℓ ∞ ; lim sup n 1 log n n k=1 |u * k | 2 < 1 B = u ∈ B ℓ ∞ ; lim sup n 1 log n n k=1 |u * k | 2 ≤ 1 .
For 1 < r < ∞, it was shown in [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF] that, setting 1 s = 1 2 + 1 max{r,2} , for every ε > 0

B ℓ r ∩ ℓ s ⊂ mon H ∞ (B ℓ r ) ⊂ B ℓ r ∩ ℓ s+ǫ . (20)
In the following we improve the previous inclusion, and show in particular that here ε = 0 is not possible. More precisely, we give necessary and sufficient conditions on (α,

β) ∈ [0, ∞[ 2 such that 1 n α log(n + 2) β n ∈ mon H ∞ (B ℓ r ) .

By Lemma 5.2 and by observing that [H

∞ (B ℓ r )] ∞ ⊂ H ∞ (B ℓ ∞ ), we get immedi- ately 1 n 1 2 • log(n + 2) θ n • B ℓ r ⊂ mon H ∞ (B ℓ ∞ ) ⊂ mon H ∞ (B ℓ r ).
Let us now prove part (1a): Assume that θ > 1 2 and 1 ≤ r ≤ 2. We shall apply Theorem 4.1, (2) with the sequence q defined by q j = j • (log( j + 2)

θ . Then for f ∈ H ∞ (ℓ r ) and u ∈ B ℓ r the conclusion follows from

1 2 + o(1) N log N • f ∞ < ∞.
Finally, we check part (2): For r = 1, (18) already proves the claim. At first we will treat the case 1 < r ≤ 2 with a probabilistic argument. Afterwards we reduce the case r ≥ 2 to the case r = 2. Let 1 < r ≤ 2. We shall apply Corollary 3.2 of [START_REF] Bayart | Maximum modulus of random polynomials[END_REF] (with p = r ). Then there is an absolute constant C ≥ 1 such that for any m, n there are (ε j ) j ∈ T J (m,n) for which We are now able to give an answer to our previously stated question: the inclusion [START_REF] Lempert | The Dolbeault complex in infinite dimensions II[END_REF] holds not true for ε = 0.

Corollary 5.6. Let 1 < r < ∞ and 1 s = 1 2 + 1 max{r,2} . Then B ℓ r ∩ ℓ s mon H ∞ (B ℓ r ).

Proof. Assume equality. Let q = k log(k + 2) k . By Theorem 5.5 this implies that the diagonal operator ℓ r → ℓ s induced by the sequence q -σ , where σ = 1 -1 min{r,2} , is well-defined and by a closed graph argument bounded. Hence

∞ k=1 q -σ k t 1 t = sup x∈B ℓ p ∞ k=1 x k q -σ k s 1 s = D q -σ : ℓ r → ℓ s < ∞ ,
where 1 s = 1 r + 1 t . Therefore q -σ ∈ ℓ t . But

∞ k=1 q -σt k = ∞ k=1 1 k log(k + 2) = ∞ , a contradiction.
Using the same technique as in the proof of (1a) in Theorem 5.5, we easily obtain the following analog of Theorem 5.3.

Corollary 5.7. Let 1 < r < ∞ and let σ = 1 -1 min{r,2} . Then

p -σ • B ℓ r ⊂ mon H ∞ (B ℓ r ) , ( 25
)
and here σ is best possible.

Lemma 3 . 3 .

 33 Let 1 ≤ r ≤ ∞ and α ∈ N n 0 with |α| = m . Then for each P ∈ P m ℓ n r we have

Theorem 4 . 1 .

 41 Let 1 ≤ r ≤ ∞ and set σ = 1 -1 min{r,2}

Lemma 4 . 3 .

 43 Let 2 < y < x and m ∈ N.

q k ≤x |c k u 1 m

 1 k | ≤ |J -(x; y)| m C m |J + (x, m; y) * | σ f ∞ . Now J + (x, m; y) * ⊂ J + (x m-, m -1) and J + (x, m; y) = for m > log x log q 1 by Lemma 4.4 and Lemma 4.3. Hence |J -(x; y)| • m

(log x) θ- 1 2

 1 loglog x , this is= x σ exp o(1) log x log log x • sup m =: exp h x,y (m) C m x -1 m y -m σ . Note that l = O(1) y (log y) θ = o(1) log x log log x ; indeed, by the definition of l l log l + 2 θ ≤ y < l + 1 log l + 3 θ ≤ l + 2 2 , hence y (log y) θ ≥ l log l + 2 θ log (l + 2) 2 θ = 2 -θ l . Differentiating h x,y (m) = m logC -1 m logxm log y , we see that it attains its maximum atM = log x log y -C ≥ log x log y ,and therefore h x,y (m) ≤ h x,y (M) = log(C ) log x log y -C = o(1) log x log log x -2 log x log y = -2 θ -1 2 + o(1) log x log log x, which proves the theorem.

Theorem 5 . 1 .

 51 Let 1 ≤ r ≤ ∞ and m ≥ 2.

1 (

 1 mr ′ ) ′ +δ ∈ mon P ( m ℓ r ), the conclusion.

1 eN=0 1 e

 11 e N <q j ≤e N +1 1 q σ j |c j ( f )u j | ≤ ∞ N=0 Nσ j∈J (e N +1 ) |c j ( f )u j | ≤ ∞ (N-1)σ e Nσ exp -2σ θ -

sup u∈B ℓ n r j ε j |j| 1 k 1 m 1 t -ε k for 1 t + 1 r = 1 2 1 j 1 (log( j 1 + 2

 11111112 u j ≤ C (n logm) σ m mσ . (23)Let now x = k -1 (log(k + 2)) -β k denote the sequence in question and assume x ∈ mon H ∞ (B ℓ r ). Then, by a closed graph argument, there exists a constant C ≥ 1, such that for every f ∈ H ∞ (B ℓ r ),j |c j ( f ) x α | ≤ C f . (24)For any n ∈ N now,u j ≤ C C (n logm) σ m mσ .by (24) and (23). Taking the m th root, we obtainn k=1 (log(k + 2)) β ≤ C C (n logm) σ m m σfor every n, m ∈ N. It now suffices to notice that with m = ⌊log n⌋ the right-hand side is asymptotically equivalent to (log n) σ and the left-hand side to (log n)1-β as n → ∞. Hence β > -σ + 1 = 1 r . Now suppose r ≥ 2 and set ξ = k -1 t (log(k +2)) -and ε > 0. Consider f ∈ H ∞ (B ℓ 2 ) and let us set g = f • D ξ , where D ξ denotes the diagonal operator ℓ r → ℓ 2 induced by ξ, which is bounded by Hölder's inequality. Thus g ∈ H ∞ (B ℓ r ). We have j |c j ( f )| f ) ξ j x j < ∞ , under the assumption that x = k -1 r -

  on Dirichlet series (see 11), and it is based on the fundamental theorem of arithmetics. In order to handle (2), we study for arbitrary finite index sets Λ of multi indices upper bounds of the unconditional basis constant of the subspace in P (

m ℓ r ) spanned by all monomials z α , α ∈ Λ. Two tools of seemingly independent interest are established. The first one is a fairly general upper estimate whenever all α ∈ Λ are m-homogeneous (i.e., |α| = m) (Theorem 3.2). The second one leads to such estimates for certain sets Λ of nonhomogeneous α ′ s, needed to apply the above technique of Konyagin and Queffélec (Theorem 4.1 and 4.2). Finally, we present our new results on sets of monomial convergence for homogeneous polynomials and bounded holomorphic functions on ℓ r (for polynomials see part (3),(4) of Theorem 5.1 and Theorem 5.3, and for holomorphic functions Theorem 5.5 with its corollaries 5.6 and 5.7).

2 r

 2 and this remains true, up to a constant factor, if we replace B ℓ n r by B ℓ n r,∞ . If we combine this with Lemma 5.2, then we find that, for r > 2, mon P

  1 2 (log(k +2)) -β k ∈ mon H ∞ (B ℓ r ) (note that ∞ (B ℓ 2 )and by our result in the case r = 2, 1 t + β + ε ≥ 1 2 for every ε > 0.

	1 t + 1 r + 1 2 = 1). Hence k(log(k + 2))

1 t + β + ε k ∈ mon H

was determined.

(2) If r = 1, then mon P ( m ℓ 1 ) = ℓ 1 .

(3

,∞

.

(4) If 1 < r < 2, then for any ε > 0, ℓ (mr ′ ) ′ -ε ⊂ monP ( m ℓ r ) ⊂ ℓ (mr ′ ) ′ ,∞ .

Several cases of this theorem are already known: the first one can be found in [START_REF] Bayart | Monomial series expansion of H p -functions and multipliers of H p -Dirichlet series[END_REF] and the second one in [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF]. The upper estimate in the third and the fourth case can also be found in [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF]. The proof of the lower estimate in the third case follows from a general technique inspired by Lemma 3.1. We need to introduce another notation. For X a Banach sequence space, R a Reinhard domain in X and F (R) a set of holomorphic functions on R, we set

[F (R)] ∞ is a set of holomorphic functions on B ℓ ∞ , and the following general result holds true.

which yields the claim.

It is now easy to deduce the lower estimate in the third case, knowing the result of part [START_REF] Balasubramanian | The Bohr inequality for ordinary Dirichlet series[END_REF]. Indeed, [P ( m X )] ∞ is contained in the set of bounded m-homogeneous polynomials on B ℓ ∞ , thus in P ( m ℓ ∞ ) by the natural extension of a bounded polynomial from B ℓ ∞ to ℓ ∞ .

The lower inclusion in ( 4) is a partial solution of a conjecture made in [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF] (see the remarks after Example 4.6 in [START_REF] Defant | Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables[END_REF]). Its proof seems less simple, and requires some preparation. Note that for r ≥ 2 we have that

which is an immediate consequence of Theorem 5.1, (3). For 1 < r < 2 we can prove this up to an ε:

we do not know whether here β = 0 is possible. Moreover, by ( 18)

if and only if β > 1. The following result collects our knowledge in the remaining cases:

In particular,

(1b) For any θ > 0 and 2 ≤ r ≤ ∞

In particular,

(2) Suppose that 1

Note that we cannot replace log(n + 2) by log(n + 1) in the previous statement. Indeed, it can be easily seen by restricting the study to the one-dimensional case that mon

Proof. We begin with part (1b): We have that θ > 0 and 2 ≤ r ≤ ∞. Then an easy argument yields 1

Proof. We proceed analogously to the proof of (1a) in Theorem 5.5 and obtain for f ∈ H ∞ (B ℓ r ) and u ∈ B ℓ r by Theorem 4.1,

Remark 5.8. Analogously to the result [START_REF] Konyagin | The translation 1 2 in the theory of Dirichlet series[END_REF] for r = ∞ and in view of Theorem 5.5, a plausible conjecture would be that for all r ≥ 2 Remark 5.9. In Theorem 5.5, the cases 1 ≤ r ≤ 2 and 2 ≤ r ≤ ∞ do not really fit for r = 2. This is due to the fact that when we apply Theorem 4.1 (2), we need that θ > 1/2. It would be nice to extend the statement of this last theorem to θ ∈ (0, 1/2].


Eliane Fustec

Eric Chauvet

Gilbert

Lignin Degradation and Humus Formation in Alluvial Soils and Sediments

HAL is

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To link to this article : URL : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC184225/ To cite this version : Fustec, Eliane and Chauvet, Eric and Gas, Gilbert Lignin Degradation and Humus Formation in Alluvial Soils andSediments. (1989) Applied andEnvironmental Microbiology, vol.55 (n°4). pp.922-926. ISSN 0099-2240 Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr


Eric Chauvet

Henri Decamps

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. To cite this version : Chauvet, Eric and Decamps, Henri Lateral Interactions in a Fluvial Landscape: The River Garonne, France. (1989) Journal of the North American Benthological Society, Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr

  This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 9640 This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 9640

	To link to this article : DOI:10.2307/1467398 To link to this article : DOI:10.2307/1467398
	URL : http://dx.doi.org/10.2307/1467398 URL : http://dx.doi.org/10.2307/1467398
	To cite this version : Chauvet, Eric and Decamps, Henri Lateral
	Interactions in a Fluvial Landscape: The River Garonne, France.
	(1989) Journal of the North American Benthological Society, vol.8
	(n°1). pp.9-17. ISSN 0887-3593


Eric Chauvet

Dynamics of seston constituants in the Ariège and

Garonne rivers (France)

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To cite this version : Chauvet, Eric and Fabre, A. Dynamics of seston constituants in the Ariège and Garonne rivers (France). (1990) Hydrobiologia, vol.192 (n°2). pp.183-190. ISSN 0018-8158 Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr


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To link to this article

  This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 9656 : DOI:10.2307/2845551 URL : http://dx.doi.org/DOI:10.2307/2845551

	To cite this version : Chauvet, Eric Aquatic Hyphomycete Distribution
	in South-Western France. (1991) Journal of Biogeography, vol.18
	(n°6). pp.699-706. ISSN 0305-0270
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