Jordy Salmon-Monviola

Chantal Gascuel

Frederick Garcia

Florent Tortrat

Marie-Odile Cordier

Véronique Masson

Ronan Trépos

Chantal Gascuel-Odoux

Marie- Odile Cordier

Simulating the effect of techniques and environmental constraints on the spatio-temporal distribution of herbicide applications and stream losses

Introduction

To reduce the impact of pesticides on human health and ecosystems, the French government introduced the 'Ecophyto 2018' plan in 2008. The main objective of this plan is to halve the current use of pesticides in France by promoting agricultural practices that use less of them. Herbicides are a particular concern since they represent one-third of the current pesticide use in France. Maize crops represent 7% of the national surface area and 10% of the current pesticide use, and 75% of the pesticides used on maize are herbicides. In regions with the highest maize production, herbicides represent most of the pesticides detected in rivers [START_REF] Aubertot | Pesticides, agriculture et environnement[END_REF]. Implementing effective measures to reduce herbicide use and losses is a challenge for agricultural and water management [START_REF] Campbell | Diffuse Pollution. An introduction to the problems and solutions[END_REF].

Many factors affect herbicide stream losses. They can be classified into four categories: (1) sitespecific factors -soil and hydrologic properties and geomorphologic characteristics of the catchment;

(2) weather factors -particularly precipitation and temperature; (3) anthropogenic factors -land-use and technical management; and (4) herbicide factors -chemical and physical properties and formulation [START_REF] Lerch | Watershed vulnerability to herbicide transport in northern Missouri and southern Iowa streams[END_REF]Müller et al., 2006). These factors have been analysed

individually (Müller et al., 2006;[START_REF] Reichenberger | Mitigation strategies to reduce pesticide inputs into ground-and surface water and their effectiveness: a review[END_REF][START_REF] Freitas | Source area effects on herbicide losses to surface waters-A case study in the Swiss Plateau[END_REF]; however, their interactions rarely have been studied. This diversity of factors demonstrates that mitigating herbicide impacts can involve regulating herbicide amounts or properties, but also involves additional constraints related to their use in space or time based on site-specific, weather, or anthropogenic factors. Such constraints on their use can concentrate or dilute herbicide applications spatially and temporally, thereby influencing herbicide losses. This paper aims to study how such constraints can modify the spatio-temporal distribution of herbicide applications and influence herbicide losses, possibly adding to the mitigating effects of simple reductions in application rates.

Finding answers to this question requires agricultural and environmental impact assessments at the meso-scale (i.e. catchments 10-50 km² in size) to account for high landscape heterogeneity.

Agricultural landscapes include farms, which contain a mosaic of fields with various slopes and soils; human infrastructures at field edges, such as grass strips, ditches, or hedgerows; and less-developed areas along streams, such as riparian wetlands. These elements now are known to either buffer or aid in pollutant transport [START_REF] Colin | Relations between triazine flux, catchment topography and distance between maize fields and the drainage network[END_REF]Leu et al., 2004a and2004b). Grass filter strips now are commonly used in regulating herbicide losses at the meso-scale. Therefore, environmental and technical constraints on herbicide applications should consider all of these natural and anthropogenic landscape elements.

Since field experiments are complex [START_REF] Reichenberger | Mitigation strategies to reduce pesticide inputs into ground-and surface water and their effectiveness: a review[END_REF], costly, and sometimes even impossible to perform, particularly at a landscape level, a modelling approach is necessary. The SACADEAU model (French acronym for "Système d'Acquisition des Connaissances pour l'Aide à la Décision sur la qualité de l'eau") is used to test the effect of herbicide application flexibility influenced by environmental and technical constraints [START_REF] Tortrat | Modèle Numérique d'Altitude (MNA) et spatialisation des transferts de surface: utilisation de structures d'arbres reliant les exutoires de parcelles et leurs surfaces contributives[END_REF][START_REF]Apprentissage symbolique à partir de données issues de simulation pour l'aide à la décision[END_REF]. This model represents biological, physical, and technical processes involved in herbicide applications and their transfer in a catchment. It combines three submodels, the first two previously described by [START_REF] Gascuel-Odoux | A decision oriented model to evaluate the effect of land use and agricultural management on herbicide pollution in stream water[END_REF]: a spatially distributed transfer model (SACADEAU-Transf), which represents biochemical and transfer processes in an agricultural landscape, a crop model, and a decision model (SACADEAU-Deci), specifically developed to test the effect of spatial and temporal constraints on herbicide applications. Technical decision processes generally are not considered in herbicide-transfer modelling [START_REF] Keating | Advances in farming systems analysis and intervention[END_REF], and simulation experiments often are based on unique dates, few herbicides, and random spatial and temporal applications [START_REF] Huber | Modeling pesticide losses with surface runoff in Germany[END_REF][START_REF] Du | Evaluation of SWAT in simulating nitrate nitrogen and atrazine fates in a watershed with tiles and potholes[END_REF]. SACADEAU-Deci fulfils the requirement for representing and analysing effects related both to environmental conditions in a catchment and agricultural constraints on one or more farms, as well as their interactions. In this model, sowing and weeding decisions are made via adaptive sequential plans that have resource and temporal constraints. The model provides agricultural interventions for sowing and weeding (date, location) and herbicide-application characteristics (substance, quantity) according to a set of predefined strategies, including weather and catchment conditions. Spatial constraints of fields are related mainly to their topography and the pattern of agricultural structures, such as farms or groups of farms. Temporal constraints are included, such as rules regarding work-time and agricultural-machine availability. Finally, weather, particularly rainfall, modifies the decision to perform agricultural activities such as sowing and weeding. Consequently, the decision model represents spatial and temporal constraints; therefore, it provides realistic herbicide-application distributions strongly determined by decisions of individual farmers and driven by technical and environmental variables. Consequently, the model predicts their effect on herbicides losses.

This study focuses on simulating maize-crop weeding and herbicide losses to streams in the months following applications to identify factors that could reduce herbicide losses to streams. It

Background

Site and data description

The study site, located in Brittany (in western France), is the 15-km² Frémeur catchment, which has a 28-km-long stream network and a drainage density of 1.65 km•km -² (Fig. 1). The slopes are moderate, with gradients of less than 5%. The landscape is made up of a medium-density bocage (a typical landscape often dedicated to animal production with fields partially surrounded by hedgerows).

The soils are silt-loams with a mean organic-matter content around 50 g•kg -1 . The soil system comprises a well-drained upland and a poorly drained bottomland. The soil overlies weathered bedrock 1-30 m deep that itself overlies fractured Brioverian schist. The physiographic setting is a temperate region with soils displaying moderate to low aggregate stability and shallow groundwater.

This study site is a few kilometres from the Naizin catchment, which has similar physiographic characteristics. This site is highly instrumented and is included in a long-term hydrological observatory (http://www.inra.fr/ore_agrhys) (Molénat et al., 2008) on which some hydrological processes have been studied in detail. The subsurface flow is quantitatively dominant (Molénat et al., 1999 and2005), and Topmodel already has been applied there successfully [START_REF] Bruneau | Sensitivity analysis to time and space resolution on an hydrological modelling based on Digital Elevation Model[END_REF][START_REF] Franks | On constraining the predictions of a distributed model: the incorporation of fuzzy estimates of saturated areas into the calibration process[END_REF]Molénat et al., 2005). The hypothesis of a water transfer time of more than one year in shallow groundwater of the upper part of the hillslope has been verified (Molénat and Gascuel-Odoux, 2002), as well a rapid and highly variable contamination of the shallow groundwater in the lower part of the hillslope (Molénat and Gascuel-Odoux, 2001). Surface flow occurs during winter as well as spring storm events, when soil surface conditions are degraded (Le [START_REF] Bissonnais | Topographic dependence of aggregate stability, overland flow and sediment transport[END_REF].

Stream water contamination is due mainly to maize herbicides and occurs in spring [START_REF] Clément | Facteurs de transfert vers les eaux de surface de quelques phytosanitaires dans le contexte agricole Breton[END_REF].

A Digital Elevation Model (DEM) of the Frémeur catchment was extracted from an elevation database for Brittany with a resolution of 20 m and was produced by stereoplotting panchromatic SPOT images to a resolution of 10 m. The parcel layer was digitised from the commune's landregistry map on a scale of 1:5000. The drainage network was extracted from the 1:25000 IGN (National Geographic Institute) map and the land-registry map. Field surveys augmented this drainage network by locating ditches, hedges, and grassed filter strips.

Agricultural land accounts for 72% of the total catchment area, the remainder being distributed among woods, wasteland, residential areas, and roads. Agricultural land is distributed as follows: maize (38%), wheat (29%), grassland (21%), and vegetables (5%); the remaining 7% includes fallow land and potatoes. The catchment contains 37 farms, of which 20 measure less than 20 ha. The catchment contains approximately 2000 parcels; in 2000, 1420 were agricultural fields including 148 in maize. Agricultural and maize fields have a mean area of 0.8 ha and 2.7 ha, respectively. This catchment, therefore, exhibits large variability in farm and field size, presenting the complex landscape mosaic common in western Europe.

We selected a 9-year period (1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002) to encompass multi-year weather variability. For each year, we focused on 1 Apr -31 Jul, a period that starts with sowing operations and herbicide applications in maize and finishes when high herbicide concentrations in stream water generally are no longer observed in the field or predicted in herbicide-transfer models. Rainfall during these periods varied moderately (mean ± SD accumulation = 215 ± 80 mm) (Fig. 2, Table 1). The frequency of rain events, characterised by the number of days with rainfall greater than 2 and 10 mm, varied from 14-41 and 2-10, respectively (Table 1). The wettest years were 1994 and 1998, while 1996 was the driest.

Past field observations provided accumulated-discharge data for the 1 Apr -31 Jul period from 1998-2002. For 1994[START_REF] Crave | The influence of the topography on time and space distribution of soil surface water content[END_REF] daily discharge was simulated by using TopModel [START_REF] Beven | A physically based variable contributive area model of basin hydrology[END_REF] (Fig. 2). The application of TopModel to this catchment for studied periods (1 Apr -31 Jul) for the 7 years where discharge measurements are available (1998)(1999)(2000)(2001)(2002)(2003)(2004), is relevant, as evidenced by Nash Efficiency criterion [START_REF] Nash | River flow forecasting through conceptual models, 1. A discussion of principles[END_REF] of 0.75. Accumulated discharge calculated for the 1 Apr -31 Jul period, which mainly depends on the rainfall during the previous months, varied from 0.03 m in 1997 to 0.16 m in 1994 (Table 1).

Weed-management practices

We identified farmers' weed-management practices in the Frémeur catchment from a 2001-2002 survey [START_REF] Tortrat | Modèle Numérique d'Altitude (MNA) et spatialisation des transferts de surface: utilisation de structures d'arbres reliant les exutoires de parcelles et leurs surfaces contributives[END_REF][START_REF] Tortrat | Modélisation orientée décision des processus de transfert par ruissellement et subsurface des herbicides dans les bassins versants agricoles[END_REF]. The 75% of farmers for whom pig production was the predominant activity usually followed simple, predefined weeding strategies recommended by technical advisers. We identified three weeding strategies: (i) pre-emergence: a single herbicide application after sowing (the most common); (ii) post-emergence: two herbicide applications at the three-and five-to seven-leaf stages of maize; and (iii) intermediate: two herbicide applications after sowing and at the five-leaf stage. Alternative strategies, such as mechanical or mixed weeding (herbicide on rows and mechanical weeding between rows), were used by 20% of the 37 farmers, who generally shared machinery within cooperative organisations. Such innovative strategies, including a zero or reduced herbicide application rate are already present despite being considered timeconsuming and technically difficult compared to a conventional chemical weeding strategy.

The herbicides used depend on the specific weeding strategy. Forty percent of the farmers on this catchment used two herbicides (Merot et al., 2009), having integrated a spatial rationale for herbicide applications: one on bottomlands (at risk for stream contamination) using less mobile and less persistent chemicals and one on uplands using chemicals considered more efficient. Fourteen different chemicals (usually applied at recommended rates) were used on this catchment, highlighting the diversity of technical advisers from agricultural organisations and private or cooperative companies who work in this area.

Herbicide application dates usually remained unknown because farmers often did not record them; thus, only general rules can be provided. Application dates are related to the weeding strategy (pre-emergence or post-emergence) and, therefore, to the stage of maize growth, which is a function of the sowing date and subsequent weather conditions. In the Frémeur catchment, sowing of each field tended to occur either in early April or late April, depending on its slope position. Fields at the bottom of the catchment had hydromorphic soil, experiencing longer periods with wet conditions than soils on the upper slopes. Therefore, to ensure soil workability, sowing and weeding operations generally were delayed on bottomland plots. Weeding operations are generally finished as quickly as possible, in regards to working constraints at the farm scale. Ultimately, herbicide application dates tended to be scattered inside different periods depending on farm and weather constraints. Since 2007, farmers are required to record herbicide application dates, but these data were neither collected nor analysed in this catchment. Hence, such data are absent in the literature, encouraging the use of a decision model to predict them. This survey revealed that techniques such as reducing the herbicide application rate, adopting spatial or temporal constraints on herbicide application, and increasing collective management of machinery or human work hours already existed in this catchment. Consequently, quantifying the efficiency of such measures and defining measures that could be emphasised to reduce herbicide losses appears relevant.

The SACADEAU model

Model overview

The SACADEAU simulation model was designed to test the effect of farmers' decisions on stream-water pollution by herbicides. The SACADEAU model combines three submodels that simulate decision-making, crop growth and pollutant transfer (Fig. 3). The latter submodel will be described briefly, as its description was published by [START_REF] Gascuel-Odoux | A decision oriented model to evaluate the effect of land use and agricultural management on herbicide pollution in stream water[END_REF]. In contrast, the decision-model is new and therefore presented in detail. Decision-model outputs constitute a portion of inputs to the transfer submodel; in contrast, there is no feedback from the transfer submodel to the decision submodel. Predicted water flows and herbicide concentrations at the catchment outlet depend upon the interaction of all submodels. Furthermore, the SACADEAU model has been included in a meta-modelling framework to identify the main factors influencing water pollution and possible mitigation recommendations, through machine-learning techniques [START_REF] Cordier | SACADEAU: a decision-aid system to improve water quality[END_REF][START_REF] Cordier | A machine learning approach for evaluating the impact of land use and management practices on streamwater pollution by pesticides[END_REF][START_REF] Trépos | A distance based approach for action recommendation[END_REF][START_REF]Apprentissage symbolique à partir de données issues de simulation pour l'aide à la décision[END_REF].

Decision submodel

The decision submodel, designed in conjunction with local agricultural experts, simulates farmers' technical decisions concerning maize crops in the spring, which include the sowing date and weed-management factors (e.g., herbicides used, dates, amounts, locations of herbicide applications) (Fig. 4). As input, the decision submodel uses the spatial distribution of maize crops on the catchment and the corresponding pre-defined sowing and weeding strategies for each of these fields. The submodel then simulates sowing and weeding activities on a daily time-step as a function of weather conditions and technical constraints.

At the field level, the submodel relies on a temporal-window approach that has been used in other decision models, such as Otelo [START_REF] Aubry | Modeling of decision-making processes for annual crop management[END_REF], Moderato [START_REF] Bergez | Improving irrigation schedules by using a crop and a decisional model[END_REF], and Déciblé [START_REF] Chatelin | DéciBlé, a software package for wheat crop management simulation[END_REF]. This notion of temporal windows, or permitted periods for carrying out an operation, represents the way farmers manage the timing of a crop operation within the overall work organisation on the farm [START_REF] Chatelin | DéciBlé, a software package for wheat crop management simulation[END_REF]. The beginning and the end of temporal windows for weeding are defined by crop growth predicted by the crop submodel (Fig. 5). When the temporal window for an operation is open, the model checks each field daily to see whether an operation's weather conditions and technical constraints are satisfied. When all conditions are fulfilled, the operation can be performed. We assumed that machine speed, availability of machines and working time, and field area determine the total duration of each operation. When the operation is finished, the operation window closes and simulation continues, waiting for the next operation window to open on the field, until the end of the simulation.

Temporal windows for sowing and weeding

In the submodel, the date of sowing influences the development of maize and soil-surface conditions, as well as dynamics of the crop and transfer submodels. For each field, at the beginning of simulation, an early-sowing (10 Apr, day-of-year 100) or late-sowing (1 May, day-of-year 120) date is defined. Weeding operations occur on one or several dates, depending upon the strategy. For example, the window for the pre-emergence strategy opens just after sowing and closes 10 days later (Fig. 5a, Table 2). If weeding cannot be performed during this period, this operation is postponed to a new window that starts at the 3-leaf stage of maize, predicted by the crop submodel (Fig. 5a). Table 2 defines the temporal windows for sowing and the two weeding strategies (pre-and post-emergence).

Parameterisation of these temporal windows was based on farmer surveys and the expertise of agricultural advisers. At the time of application, the submodel chooses herbicide types and doses from a database based on each farmer's weeding strategy.

Environmental and technical conditions

We assumed that farmers do not sow or apply herbicides on rainy days and that machines cannot work on a field when its soil is too wet to support them. Thus, in the submodel, operations can be performed only if daily rainfall is less than 2 mm and at least 2 days have passed since it last exceeded 2 mm (Table 2). The submodel also contains a daily working-time limit (T) for sowing and weeding operations per farmer. To this limit, overtime (ε T ) can be added to finish an operation on a given field in the same day. We assumed that only one machine performs each operation on a field, at rates of 1 and 4 ha per hour for sowing and weeding, respectively (Table 2).

Spatial organisation

The submodel allows organisation of machines at three spatial scales (farm, farm group, and catchment) to determine their use on individual fields. At the individual-farm level, each farm has its own set of machinery and fields that do not change over time (the current scenario of the Frémeur catchment). At the farm-group scale, a farm that has completed its sowing or weeding operations may share machinery with other farms in the group. At the catchment scale, a downhill-slope index [START_REF] Crave | The influence of the topography on time and space distribution of soil surface water content[END_REF]Merot et al., 2003), slightly modified from the Beven index [START_REF] Beven | A physically based variable contributive area model of basin hydrology[END_REF], is used to classify each field in the catchment as well-drained (uplands) or wet (bottomlands). This index is a good predictor of hydromorphic soils (Merot et al., 2003). Since soil workability is linked to soil water content [START_REF] Rounsevell | Climate Change and Agricultural Soils: Impacts and Adaptation[END_REF] and this latter variable is controlled by topography [START_REF] Crave | The influence of the topography on time and space distribution of soil surface water content[END_REF], we assumed in the submodels that farmers first work on upland fields of the catchment and then move to bottomlands. The submodel also can use this index to fix the sowing date of each field. Including these three spatial scales in the decision submodel provided the ability to simulate the spatial and temporal distribution of agricultural operations over the catchment in a more realistic manner.

Crop submodel

The only prediction required from the crop submodel is the number of leaves on maize, which determines the starting dates of temporal windows for weeding operations. Leaf number is predicted from cumulative degree days after sowing [START_REF] Hammer | Modelling genotypic and environmental control of leaf area dynamics in grain sorghum. I. Whole plant level[END_REF]) with a linear equation (i.e., 100, 200, and 300°C to reach 3, 5, and 7 leaves, respectively).

Herbicide-transfer submodel

The transfer model, detailed in [START_REF] Gascuel-Odoux | A decision oriented model to evaluate the effect of land use and agricultural management on herbicide pollution in stream water[END_REF], simulates surface and subsurface flow of herbicide in water, from its application on a field to the outlet of the catchment. The model separately calculates the discharge at the outlet of the catchment and the amount of pesticide transferred by surface and subsurface flow. Here, discharge is considered a global dilution factor affecting the amount of pesticide mobilised in the surface and subsurface flow.

Surface flows of water and pesticide are aggregated at the catchment level using a tree structure linking the plot outlets and their contributing areas. This spatial representation is based on a spatial object-based modelling approach detailed elsewhere [START_REF] Tortrat | Modèle Numérique d'Altitude (MNA) et spatialisation des transferts de surface: utilisation de structures d'arbres reliant les exutoires de parcelles et leurs surfaces contributives[END_REF][START_REF] Aurousseau | A plot drainage network as a conceptual tool for the spatial representation of surface flow pathways in agricultural catchments[END_REF]. It allows the upslope surface flow to infiltrate in downslope plots or linear networks. If present, these linear networks, such as hedges or ditches, modify both the flow direction and the location of field outlets, altering this tree structure. According to this spatial model, only 125 of 148 maize fields on the Frémeur catchment were connected to the stream; the other fields were considered total sinks. Surface flow is controlled either by soil surface sealing on the whole catchment or saturated conditions in bottomlands. The latter is related to the subsurface flow submodel. The surface flow controlled by soil surface sealing is estimated using the concepts and tools developed in the Stream model [START_REF] Cerdan | Incorporating soil surface crusting processes in an expert-based runoff model: sealing and transfer by runoff and erosion related to agricultural management[END_REF], an expert-based runoff model using expert rules in the form of matching tables characterising agricultural fields according to soil surface conditions (roughness, soil surface sealing, crop cover) to determine the soil infiltration capacity. This model has been successfully used in a variety of conditions [START_REF] Evrard | Reliability of an expert-based runoff and erosion model: Application of STREAM to different environments[END_REF].

The subsurface flow model aims to delineate saturated surfaces and estimate the amount of pesticide reaching the groundwater. This is carried out by estimating the storage of pesticide at the soil surface and a transfer coefficient depending on the depth of the water table. The depth of the water table is estimated only on the lower part of the hillslope, since water transfer time in shallow groundwater is estimated to be more than one year for the upper part of the hillslope, which is larger than duration of the studied simulations (4-5 months). Topmodel is used to calculate a mean saturation deficit at each point of the catchment, depending on a topographic index, which can be linked to the water table depth. This submodel provides a daily estimate of the depth of the water table per plot, grouped into different classes over the catchment. Finally, subsurface quantities of herbicide coming from these classes are aggregated into a single reservoir of constant volume that is drained according to a constant drainage coefficient linearly dependent on the catchment's mean water table.

Water and pesticide transfer are coupled in a unique way for the surface and subsurface flow. The initial herbicide concentration in the soil is calculated assuming a complete, rapid, and reversible mixing area between soil and herbicides. Herbicide degradation is predicted using a first-order kinetic equation with a standard half-life parameter. Before each rainfall event, the herbicide quantity in soil is calculated taking into account degradation and surface and subsurface transfer processes since the previous rainfall event. A constant exchange coefficient between soil and water and a fixed area of exchange is used to calculate the herbicide concentration of surface flow. Finally, the amount of herbicide exported by surface and subsurface flow during rainy days is predicted, and the new amount of herbicide stored in soil is calculated. This model has been calibrated on the Frémeur catchment and is partly validated by observations [START_REF] Gascuel-Odoux | A decision oriented model to evaluate the effect of land use and agricultural management on herbicide pollution in stream water[END_REF].

Model output

The SACADEAU model predicts two variables at the catchment outlet that are spatially and temporally aggregated:

-Weeding Day Accumulation (WDA) is the difference between the last day (LastWeedingDay) and the first day (FirstWeedingDay) of herbicide application on the entire catchment: WDA = (LastWeedingDay -FirstWeedingDay + 1)

(1)

-T-ratio is the ratio of the total amount of herbicides transferred by surface and subsurface routes to the catchment outlet (herbicide_output) over the total amount of herbicides applied on the catchment (herbicide_input):

T-ratio = ∑ ∑ t t input herbicide_ ouput herbicide_ * 100
(2)

WDA summarises the effect of different environmental and technical constraints on the temporal distribution of weeding operation, while T-ratio summarises the effect of interactions between the spatial and temporal distribution of herbicide applications and the spatial structure of the catchment on the herbicide-transfer rate. Since T-ratio is the output variable of the transfer submodel, the accuracy and robustness of its predictions have been discussed previously [START_REF] Gascuel-Odoux | A decision oriented model to evaluate the effect of land use and agricultural management on herbicide pollution in stream water[END_REF].

Simulation experiments

Protocol of simulations

Simulation experiments tested temporal factors related to availability of machines and working time; spatial factors created by the spatial structure of farms, farm groups, or the catchment; and factors related to herbicide dose and proportion of fields treated (Table 3). All simulations were conducted using nine years of weather records (Fig. 2, Table 3). To simplify the interpretation of results, only one weeding strategy (pre-emergence) and one herbicide (dimethenamid) were chosen.

We set the properties of dimethenamid, a chloroacetamid, as follows: sorption partition coefficients between water and organic carbon, Koc = 260 cm 3. .g -1 •; standard half-life, DT50 = 20 days; application rate of 1.6 L ha -1 (active ingredient 1440 g ha -1 , dimethenamid).

Temporal factors: availability of machines and working time

Experiment 1.1 represented the baseline simulation with no constraints. When conditions allowed working on a field, the operation was performed regardless of the predicted availability of machines or working time. Subsequently, five simulation experiments were performed to study the effect of machine availability (Table 3). Working time was fixed at 8 hours, with a possible overtime of one hour. Each machine was considered as operated by one person, which corresponded to 8 hours of work. For these experiments, an early sowing date was set for fields at higher elevations of the catchment (80% of the fields) and a later date for fields at lower elevations. In experiment 1.2, one machine was allocated per field. In experiment 1.3, one, two or four machines were allocated per farm.

In experiment 1.4, one, two or four machines were allocated for the entire catchment (all farms combined). In experiment 1.5, one, two or four machines were allocated for the entire catchment, but machines also managed at farm level. Thus, when a farm's field needed to be worked and a machine was not available on the farm itself, one was allocated if available from the catchment machine pool.

This assumption allowed several machines to be allocated to the same farm at the same time. Once the sowing or weeding operation was finished, the machine returned to the catchment pool. Finally, experiment 1.6 was a modification of experiment 1.5 in which the spatial location of fields determined the allocation of machines. When all upland fields (80% of the fields in the catchment) of a farm were sown or weeded, the machine returned to the catchment machine pool.

One experiment was performed to study the effect of farmer working-time (experiment 2.1, Table 3). It considered a maximum working time per day of 4, 6, or 8 hours, with possible overtime of one hour. The minimum working time was set at 4 hours to simulate the fact that farmers do not dedicate all their time to maize-related operations. For these experiments, two dates of sowing and one machine per field were assumed.

Spatio-temporal distribution of sowing operations

Additional simulation experiments tested different sowing dates as a function of the topographic position of fields (Table 3). For these experiments, one machine per field and 8 hours of working time were assumed. Four scenarios were tested: i) all fields with an early sowing date (experiment 3.1); ii) all fields with a late sowing date (experiment 3.2); iii) 80% of the fields with an early sowing date and 20% of the fields with a late sowing date, based on the topographic index as spatial criterion (experiment 3.3); and iv) 150 replicate simulations of random allocations of the late sowing date for 20% of the catchment fields regardless of location to introduce more variability into the previous scenario (experiment 3.4). This last experiment allows us to analyse the effect of different spatial allocation.

Percentage of plots treated and herbicide dose

A final set of five simulation experiments tested different strategies of herbicide reduction, either by reducing the number of fields treated or the herbicide dose (Table 3). For these experiments, two dates of sowing, one machine per field, and 8 hours of working time were assumed. The first experiment varied the percentage of treated fields from 30-100% (experiment 4.1). Then, fixing the percentage of treated fields at 50% (62 plots), we introduced three types of random field selection, performing 150 replicate simulations per year to avoid the effect of specific scenarios on the results.

The remaining four experiments simulated ii) random selection among all fields in the catchment (experiment 4.2); iii) random selection of 25 bottomland fields and 37 upland fields using a spatial criterion (experiment 4.3); iv) random selection of 62 upland fields (experiment 4.4); and v) variation in herbicide dose on all plots from 15-100% (experiment 4.5).

Results

Temporal factors: availability of machines and working time

Depending upon the year, predicted WDA values varied from 2-75 days (Fig. 6a and6b), while predicted T-ratios varied from 0.1-2.2% among all experiments and years (Fig. 6c and6d). The low value and variability of predicted T-ratios indicate that a small proportion of herbicides were transferred to the stream, regardless of weather conditions. The highest values of the T-ratio in all experiments corresponded to the two wettest years (1994 and 1998).

Predicted WDA values generally presented the same pattern, marked by low values in 1998, 1999, and 2000 and high values in 1995, 1997, and 2002. The T-ratio and WDA values showed inverse trends: when WDA was short (4 days) the T-ratio was high (1998) and, conversely, when WDA was long (about 25 days), the T-ratio was low (1995-1997, 2001, and 2002). But the variability of the T-ratio cannot simply be explained by a single variable (WDA), as was seen in 2000, when both the predicted WDA and T-Ratio were low (Fig. 6a and6c).

Three groups of experiments with similar responses could be distinguished regarding WDA.

The first corresponded to experiments with any number of machines at the farm or catchment scale, without any consideration of farm relations (experiments 1.1-1.4); among all years, predicted WDA values remained below 20 days (Fig. 6a). The second group corresponded to experiments with only one or two machines per catchment, but considered the spatial constraints of farm structure (experiment 1.5) and catchment topography (experiment 1.6 with 2 machines per catchment), for which predicted WDA varied among years from 40-75 days (Fig. 6b). The third group assuming four machines available with the same constraints (experiments 1.5 and 1.6) presented intermediate results (Fig. 6b). The values of the T-ratio could not be split clearly between these groups because of different trends from year to year. Therefore, the effect of machine availability on the T-ratio initially depended on the year. This was particularly well observed in experiment 1.5, with one machine available for the entire catchment. In this case, the T-ratio was either the lowest (1994, 1996, 1998, and 2002) or the highest (1995, 1997, 1999, and 2000) (Fig. 6d). Moreover collective machine management at the catchment scale that also considers the farm scale may reduce herbicide transfer significantly during some years by temporarily increasing in time the distribution of weeding operations like in 1994 (Fig. 6b and6d).

The availability of working time as tested here (experiment 2.1) had no effect on predicted WDA or T-ratio values, which were similar to those in the baseline (experiment 1.1). Working time did not constrain the spatial distribution of weeding operations and therefore had no effect on the Tratio.

Spatio-temporal distribution of sowing operations

Predicted WDA was short for both early and late sowing (Fig. 7a). As expected, a scenario with 80% early sowing and 20% late sowing (experiment 3.3) greatly extend this duration. However, this experiment did not give the lowest T-ratio.

The effect of the sowing date on the T-ratio also depended on the year (Fig. 7a). From 1997-2000, this factor has no effect on the T-ratio, whereas the effect was large for 1994, 2001, and 2002.

Early sowing for all fields (experiment 3.1) gave the lowest T-ratio, except for 2001. The results for the scenario with 80% early sowing and 20% late sowing (experiment 3.3) provided a little higher Tratio than the experiment 3.1. Considering this latest scenario (experiment 3.3), the results were relatively similar when date of sowing of fields were selected randomly (experiment 3.4) (Fig. 7b), except in 1994 and 2002 for which T-ratio was lower, and 2001 for which it was higher. These results

show that considering different sowing dates, with (Fig. 7a) or without spatial criteria (Fig. 7b), did not consistently lead to a lower T-ratio except for a particularly rainy year like 1994 when early sowing can be promoted.

Percentage and allocation of fields treated and herbicide dose

The T-ratio decreased as the number of weeded fields decreased for all years (experiment 4.1), except in 1997, when it was always near zero. The range of the decrease depends on the year (Fig. 8a).

It decreased from 2 to 0.6% and from 1.5 to 0.7%, in 1994 and 1998, respectively (Fig. 8a). A slight decrease in the percentage of fields weeded from 100 to 90% led to a significant decrease of the T-ratio, particularly when the T-ratio was high, as in 1994. Lastly, reducing the percentage of fields weeded decreased the T-ratio more than reducing the application rate (Fig. 8a). Indeed, the results of experiment 4.1 with 100 % of fields weeded were similar to those of experiment 4.5, regardless of the herbicide dose. In comparison to experiment 4.2, when a larger percentage of treated fields were randomly selected from bottomland (experiment 4.3), T-ratio increased (Figs. 8b and8c). When herbicide applications were limited to upland fields (experiment 4.4), predicted T-ratios were lower than 1% (Fig. 8d), and the range of variation was lower than those of the other two spatial scenarios (Figs. 8b and8c).

Discussion

Utility, evaluation and improvement of the model

Developing and combining a decision model with crop and transfer models constitutes the uniqueness of this modelling approach; it shows that herbicide stream-water contamination is the result of a conjunction of environmental conditions, such as weather and soil patterns, and technical constraints, such as machine availability.

The relevance of a model comes from the innovative measures that it can suggest or from the support it can provide to stakeholders regarding mitigation measures. From this point of view, we can emphasise the unexpected effect of machine availability on predicted herbicide losses in our simulations, which requires further investigation. We also highlight the comparison of the effects of different herbicide-reduction strategies, which suggest the potential for reducing herbicide losses by implementing spatial constraints on herbicide applications.

The validation of such a model from observations is challenging. A rigorous validation process would require detailed data on the spatial and temporal distribution of the quantities of herbicides applied over the catchment. These data are not available at the present time, especially not for a long time-series, but they should be in the future, now that recent regulations require recording quantities of herbicide applied. In contrast, the elaboration of the model and the distribution of simulated sowing and weeding dates were discussed with local agricultural experts, who deemed them coherent.

A sensitivity analysis of the parameters was performed on the allocation of sowing and weeding dates.

The number of replicate simulations (150 per year) (Fig. 7b and8b,c,d) and the length of the weather series (nine years) covers a large range of environmental and technical conditions and yielded a predicted T-ratio that ranged from 0.1-3.5%. The observed T-ratio ranges from 0.1 to 0.6 % for six years [START_REF] Clément | Facteurs de transfert vers les eaux de surface de quelques phytosanitaires dans le contexte agricole Breton[END_REF] and therefore is included in the range of simulated T-ratio. The larger range of T-ratio is explained by a larger range of simulated conditions. Predictions were made assuming an average herbicide decay-rate (DT50), but it would be useful to test a wider range of values of this property, since they may influence herbicide losses greatly. Further experiments could also be done to test the effect of spatial and temporal distribution of agricultural activities. Especially, the spatial rule 'farmers first work on upland fields of the catchment, then move to bottomlands' could be relaxed or changed by analyzing its effects on T-ratio and WDA values. This rule is adapted to the studied environmental conditions where the bottom lands are longer wet than uplands, but could be relaxed in other environmental conditions.

The model could be improved by including other processes such as the pre-sowing operations of ploughing or seed-bed preparation. These pre-sowing operations will influence the timing of operations that follow, depending on soil type and weather [START_REF] Leenhardt | Estimating the spatial and temporal distribution of sowing dates for regional water management[END_REF], and influence water infiltration and surface runoff. Also, strategies based on observations of weed encroachment could be taken into account to adapt weeding operations. This improvement would require a crop model that represents weed growth and crop-weed competition. Adding equations to quantify yield effects of the different strategies also could help test their acceptability to farmers.

Functions to adapt the herbicide dose to weather conditions also could be added. Because of the modular structure of SACADEAU, it has few limits to the inclusion and the test of different or more sophisticated models. Lastly, the output variables studied could be improved, in particular WDA, which describes the duration of weeding operations but not the effective number of working days for weeding or the spatial distribution of weeding operations. To estimate spatial distribution, an equation that calculates the mean distance of weeded plots to the stream over time could be proposed.

Interactions between agricultural practices and weather conditions

The effect of weather on the WDA and the T-ratio is obvious but complex due to interactions between weather conditions and the spatial distribution of sowing and weeding operations. The effect of rainfall amount and frequency during each year is high. For the rainiest years of 1994 and 1998, the T-ratio and its variability are highest due to large individual rainfall events and rainfall frequency.

The date of one rainfall event can have a large effect on the WDA and the T-ratio. For example, in the case of machine availability experiments, in simulation experiment 1.6 in 1998 (a rainy year), the WDA with one machine per catchment had a lower predicted WDA (18 days, days-of-year 150-167) than that with two machines per catchment (65 days, days-of-year 103-167). Having one fewer machine on the catchment moved sowing dates to a rainier period in which pre-emergence weeding became impossible, delaying the entire weeding program to the 3-leaf stage.

The distribution of rain also could have a significant effect on the WDA and the T-ratio, as

shown when comparing the results experiments 3.1 (early sowing) and 3.2 (late sowing) in the two rainiest years (1994 and 1998). The predicted early-sowing T-ratio was half that of the late-sowing Tratio in 1994 but not in 1998 (Fig. 7a), despite similar rainfall amounts (Table 2). Analysis of the simulations shows that rainfall events in early spring 1998 concentrated weeding operations around day-of-year 122 regardless of the sowing date. Conversely, the absence of rainfall in early spring 1994 separated weeding operations into two periods according to the sowing strategy. Consequently, in 1994 predicted herbicide concentrations in the soil were lower after early sowing than late sowing because they had approximately 20 additional days to degrade. The low predicted T-ratio for the early sowing date in all 9 years shows that this date best limited herbicide transfer; however, this effect can be modified by spring rainfall distribution.

Weather factors had smaller effects on the T-ratio, with low intra-and inter-year variability when herbicide applications were located on upland fields (Fig. 8d). Analysis of the relation between predicted WDA and T-ratio for all experiments and all years (Fig. 9) shows that these variables were partially correlated in years of exceptional rainfall (1994 and 1998). In 1998, T-ratio increased as WDA increased, whereas in 1994, for WDA higher than 40 days, T-ratio decreased as WDA increased (Fig. 9). In other years, the predicted T-ratio ranged from 0.2-1% regardless of WDA.

Finally, rainfall can delay herbicide applications, temporally concentrating them on a catchment, and increase herbicide transfer to streams by increasing runoff; both of these rainfall effects can increase the herbicide transfer ratio. A better investigation of the influence of rainfall could be achieved by using simulated weather in the model that specifies rainfall amount and frequencies, and by improving modelling at short timescale. The surface runoff model operates at hourly timescale, which does not allow the model to simulate correctly runoff and herbicide transfer when short and intense rainfall events occur at minutes to few hours scale. Not taking such flushing events into account may lead to underestimate the water and herbicide transfer.

Strategies to reduce herbicide pollution

Despite strong interactions between agricultural activities and weather conditions, and high inter-annual variability of T-ratio, which weeding strategies can be recommended? The results of the numeric experiments have been aggregated over the nine chosen years to compare them (Fig. 10). An experiment is more effective as the median and the variability are low, i.e. effective whatever the climatic conditions. From this criterion, experiments which reduce proportion of plots treated (experiment 4.1), especially by avoiding herbicide application bottomland fields of the catchment (experiment 4.4), are particularly effective. This mitigation measure must be undertaken collectively at the catchment scale because it would not imply the same constraints on all farms.

Among all experiments, variations in working time had no effect because the 4-hours minimum was sufficient to perform the operations due to small size and low number of plots per farm, small farms sizes and high machine-throughput rates. As this variable seems to be context dependant, the application of the model to other contexts (larger plots, more plots, and more farms) could be interesting especially to test the sensitivity of this variable to the context, and analyze the effect of higher up to extreme of working time when conditions are optimal for farmers.

Similarly, machine management simultaneously at the farm and catchment scales always was sufficient to apply herbicide to all catchment fields (Fig. 6). As for the T-ratio, we can say that working time and machine availability are often over-dimensioned in simulations. A simulation considering that only 1-2 machines per catchment were able to distribute weeding operations over time and halve the T-ratio from 2.2 to 1% in 1994 and 1.6 to 0.8% in 1998 (experiment 1.5, Fig. 6d).

Collective machine management and constraints on machine availability could be considered as potential factors to reduce water contamination at this stage of an exploratory approach. These scenarios would have to be tested with a more complex crop model to evaluate the effects of such measures on crop development, yield, and weed management and evaluate their acceptability by farmers and to analyse the variability regarding climatic conditions.

Conclusion

The decision submodel was developed to simulate decisions of farmers for sowing and weeding activities that take into account environmental constraints such as weather and slope position of the plots and technical constraints such as the availability of machines and farmers' working time. These operations are performed during temporal windows if certain conditions are fulfilled. The decision submodel distributes a given agricultural operation over time and space more realistically than random or unique methods commonly used in numerical simulations.

Simulation results show that herbicide transfer is not only the effect of the quantity of herbicides applied, but of technical and environmental factors that interact to concentrate or spread herbicide applications over time and space. Herbicide transfer depends greatly on annual weather conditions.

Collective machine management and an early sowing date can decrease herbicide transfer, but their effects vary according to weather conditions. Nonetheless, these practices could be promoted more frequently. Spatial strategies that decrease the number of fields in the catchment on which herbicides are applied are always effective, particularly when herbicide applications occur only on upland fields. 3 for simulation experiments abbreviations. See Table 3 for simulation experiments abbreviations. See Table 3 for simulation experiments abbreviations. Table 3 for simulation experiments abbreviations.

  addresses two questions: (1) To what extent do technical constraints such as availability of machinery or work time, or environmental constraints such as topographic position influence herbicide losses? (2) Can delineating the location or period of sowing or weeding based on environmental or technical factors further reduce herbicide losses?

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Rainfall (mm/day) and discharge (m/day) on the study catchment from 1 Apr to 31 Jul for

Figure 3 .

 3 Figure 3. General SACADEAU model diagram. The T-ratio equals the ratio between predicted

Figure 4 .

 4 Figure 4. Decision-model diagram. ε Rainfall = rainfall threshold; t RainfallEvent = day with rainfall higher than ε Rainfall Figure 5. Temporal diagram of decisions for (a) pre-emergence and (b) post-emergence weeding

Figure 6 .

 6 Figure 6. Predicted T-ratio and WDA from 1994-2002 for simulation experiments 1.1 to 1.6. (mach =

Figure 7

 7 Figure 7. a) predicted T-ratio and WDA from 1994-2002 for simulation experiment 3.1-3.3 (3.1: all

Figure 8

 8 Figure 8. a) predicted T-ratio from 1994-2002 when the percentage of weeded fields decreased

Figure 9 .

 9 Figure 9. Comparison of predicted WDA (Weeding Day Accumulation) and T-ratio for all simulation

Figure 10 .

 10 Figure 10. Standard boxplots of predicted T-ratio for all experiments with all years aggregated. See

  Finally, our simulations indicate that modifying the spatio-temporal distribution of herbicide applications by considering environmental and technical constraints does not automatically decrease herbicide transfer rates, as assumed. The effect of a higher flexibility in time and space in herbicide applications appears to depend strongly on weather conditions, generally becoming more effective during rainy years.Leu, C.M., Singer, H., Stamm, Ch., Müller, S.R., Schawarzenbach, R.P., 2004a. Simultaneous assessment of sources, processes, and factors influencing herbicide losses to surface waters in small agricultural catchment.Environ. Sci. Technol. 38, 3827-3834. Leu, C.M., Singer, H., Stamm, Ch., Müller, S.R., Schawarzenbach, R.P., 2004b. Variability of herbicide losses from 13 fields to surface waters within a small catchment after a controlled herbicide application. Environ. Sci. Technol. 38, 3835-3841. Merot, P., Aurousseau, P., Gascuel-Odoux, C., Durand P., 2009. Innovative assessment tools to Merot, P., Squividant, H., Aurousseau, P., Hefting, M., Burt, T.P., Maitre, V., Kruk, M., Butturini, M., Thenail, C., Viaud, V., 2003. Testing a climato-topographic index for predicting wetlands distribution along an European climate gradient. Ecol. Model. 162, 51-71. Molénat, J., Davy, P., Gascuel-Odoux, C., Durand, P., 1999. Study of three subsurface hydrologic systems based on spectral and co-spectral analysis of time series. J. Hydrol. 222, 152-164. Molénat, J., Gascuel-Odoux, C., 2001. Role of shallow groundwater in nitrate and herbicides transport in the Kervidy agricultural catchment (Brittany, France). IAHS Pub. 269, 347-351. Molénat, J., Gascuel-Odoux, C., 2002. Modelling flow and nitrate transport in groundwater for the prediction of water travel times and of consequences of land use evolution on water quality.Hydrol.Process. 16, 479-492. Molénat, J., Gascuel-Odoux, C., Davy, P., Durand, P., 2005. How to model shallow water-table depth variations: the case of the Kervidy-Naizin catchment, France. Hydrol.Process. 19, 901-920. Molénat, J., Gascuel-Odoux, C., Ruiz, L., Gruau, G., 2008. Role of water table dynamics on stream nitrate export and concentration in agricultural headwater catchment (France). J.Hydrol. 348, 363- 378. Müller, K., Stenger, R., Rahman, A., 2006. Herbicide loss in surface runoff from a pastoral hillslope in the Pukemanga catchment (New Zealand): Role of pre-event soil water content. Agric. Ecosyst.

	improve	water	quality	and	watershed	management	in	farming	areas.
	Integrated Environmental Assessment and Management, 5, 158-166.

Environ. 112,[381][382][383][384][385][386][387][388][389][390]

Acknowledgments

We thank the Chamber of Agriculture of Brittany (France) for providing the data. This work was financed under the project "Sacadeau" set up within the framework of INRA-CIRAD Transverse Action "Decision-making aid: how to link together knowledge and action in agriculture, agro-food industry and rural areas" and the project "APPEAU" within the ANR "Agriculture and sustainable development". The authors also thank Dr M Corson for his constructive remarks and for reviewing the English.


Alessandra Bernardi
email: alessandra.bernardi@inria.fr

Iacopo Carusotto
email: carusott@science.unitn.it

Algebraic Geometry tools for the study of entanglement: an application to spin squeezed states

Keywords: 03, 65, Ud Quantum entanglement -02, 10, -v Algebraic geometry -42, 50, Dv Squeezed states

A short review of Algebraic Geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics. Examples of application to assemblies of indistinguishable two-level bosonic atoms are discussed using modern formulations of the classical Sylvester's algorithm for the decomposition of homogeneous polynomials in two variables. In particular, the symmetric rank and symmetric border rank of spin squeezed states is calculated as well as their Schrödinger-cat-like decomposition as the sum of macroscopically different coherent spin states; Fock states provide an example of states for which the symmetric rank and the symmetric border rank are different.

Introduction

The mathematical research on tensor decomposition is generally considered to have started out in the late XIX century with the pioneering work of J. J. Sylvester on the decomposition of a homogeneous polynomial in two variables as the sum of powers of linear forms. Still now, this kind of questions are attracting a strong interest from the algebraic geometry community and they have been related to the problem of decomposing a multi-dimensional, multi-index tensor as the sum of simpler, rank-1 objects; the minimum number of elements of such a decomposition is called the rank of the tensor. Given the isomorphism between homogeneous polynomials and totally symmetric tensors, Sylvester's result is a first example in this program. In the last years, the algebraic geometry tools for tensor decomposition are starting being applied in a number of application fields, ranging from Signal Processing [START_REF]Mathematics in Signal Processing V[END_REF][START_REF] Comon | Signal Processing[END_REF][START_REF] Bernardi | ISAAC[END_REF], Telecommunications [START_REF] Sidiropoulos | [END_REF]5] to Phylogenetics [6][7][8], Chemometrics [9] and Algebraic Statistics [START_REF] Sturmfels | sue: 10 years of BADMath[END_REF]11], Geometric complexity theory [12][13][START_REF] Buergisser | Linear Algebra and Applications[END_REF], Arithmetic complexity [15][START_REF] Bini | [END_REF][17][18], Data Analysis (e.g. Independent Component Analysis [START_REF] Comon | Higher-Order Statistics[END_REF][START_REF] Cardoso | Proc. of the IEEE[END_REF][21] ).

In physics, the Hilbert space of a composite quantum system is the tensor product of the Hilbert spaces of the constituent sub-systems, e.g. the individual particles forming a quantum gas or the entangled, yet physically separated sub-systems of quantum information processing devices. Also in this context, it is often of interest to expand a given quantum state as the sum of factorizable states, i.e. tensor products of pure states of each sub-system. The minimum number of factorizable states that have to be summed to recover a given state is called the Schmidt rank of the state and is often mentioned as a possible measure of entanglement [22,23]. In many cases, the decomposition of the state has itself a physical interest, as it highlights an underlying Schrödinger cat structure as a linear superposition of macroscopically different states.

In the present paper, we investigate the application of a algebraic geometry techniques to the calculation of the rank and of the decomposition of quantum states of actual physical interest of an assembly of N indistinguishable two-level atoms, in particular spin squeezed states and Fock states. On one hand, two level atoms are amenable to a description in terms of bivariate homogeneous polynomials that can be tackled using Sylvester's algorithm and its modern extensions [24][25][26]. On the other hand, spin squeezed states have been proposed as a practical tool to overcome the shot noise limit in precision interferometrical measurements [27][28][29]: experimental demonstrations of this idea have been reported using either the internal atomic level structure in a cloud of thermal [30] or Bosecondensed [31] atoms or the atomic center-of-mass motion in a double well geometry [32].

The article is organized as follows: the relation between quantum states, tensors and polynomials is reviewed in Sec.2. A general introduction to the concept of symmetric rank and symmetric border rank is given in Sec.3. The basic definitions are given in Sec.3.1, together with a physical interpretation in the second quantization formalism: for symmetric (skewsymmetric) tensors describing a Bose (Fermi) system, the decomposition of the physical state is made over Hartree (Hartree-Fock) states. A modern formulation of Sylvester's algorithm of [25] is presented in Sec.3.2.

Sec.4 is devoted to the application of these concepts to specific physical states, namely spin squeezed states and Fock states. The calculation of the symmetric rank and border rank of spin squeezed states is presented in Sec.4.1: the symmetric rank and symmetric border rank turn out to be always equal for this class of states, but interesting features are found whenever the squeezing parameter is a rational multiple of 2π, µ = 2πp/q. For these values, the symmetric rank is lower than its generic value; as originally predicted in [33], the state can be decomposed as the sum of q macroscopically different coherent spin states pointing on equispaced directions along the horizontal plane orthogonal to the quantization axis. Sec.4.2 is devoted to Fock states with a well defined occupation number of the two states, for which the symmetric rank and symmetric border rank turn out to be different. Explicit decompositions are given as an exact sum or as the limit of sums of coherent spin states aligned on the horizontal plane.

Conclusions and perspectives are summarized in Sec. 5.

Quantum states, tensors and polynomials

We consider a system of N quantum particles. The Hilbert space H 1 for each individual atom is assumed to have dimension n. The Hilbert space of the N -body system is obtained as the tensor product of N copies of the n-dimensional single particle Hilbert space H 1 . In the case of indistinguishable bosonic (i.e. with integer spin) particles, the indistinguishability postulate of quantum mechanics imposes that only the totally symmetric states under particle exchange are physically relevant, which amounts to restricting our attention to the subspace H s = S N H 1 ⊂ N H 1 of completely symmetric tensors.

Using the {e 1 , . . . , e n } orthonormal basis of H 1 , a symmetric tensor t can be written as

t = 1≤i j∈{1...N } ≤n a i1,...,i N e i1 ⊗ . . . ⊗ e i N (1)
where the coefficients tensor a (i.e. the many-body wavefunction in the {e 1 , . . . , e n } basis) is symmetric under exchange, a ...ij ...i j ... = a ...i j ...ij ... for all i j , i j = 1, . . . , n.

The standard isomorphism (i.e. biunivocal linear map) between symmetric tensors and homogeneous polynomials associates the tensor t ∈ S N H 1 defined in (1) to the following homogeneous polynomial p(x 1 , . . . , x n ) of degree N ,

p(x 1 , . . . , x N ) = 1≤i j∈{1...N } ≤n a i1,...,i N x i1 . . . x i N . (2)
A related isomorphism between physical states and polynomials naturally arises within the second quantization description of many-body states in terms of the usual creation operators â † i for an atom in the i state. Each monomial

x n1 1 • • • x n N
N of degree N is associated to a (unnormalized) physical state with a well defined number of particles n i in each single particle state i = 1, . . . , n and total number of particles

d i=1 n i = N , x n1 1 • • • x n N N ←→ (â † 1 ) n1 • • • (â † n ) nn |vac . (3)
It is immediate to see that this isomorphism can be extended by linearity to the whole space of polynomials and to the whole many-body Hilbert space H s . With the proper normalization factor, the isomorphism between polynomials and physical manyparticle states induced in second quantization by ( 3) is fully equivalent to the standard one of the mathematical literature summarized by ( 1) and ( 2). An analogous isomorphism between physical states and Grassmann-generalized polynomials in anti-commuting variables can be defined for systems of fermionic (i.e. with half-integer spin) particles for which the many-body wavefunction is totally antisymmetric under exchange. Such states correspond to skewsymmetric tensors and, within second quantization, can be written in the form (3) in terms of creation operators â † i which now anti-commute.

Rank and Border rank

Definition and physical interpretation

For a given symmetric tensor t ∈ S N H 1 , its symmetric rank sr(t) is defined as the minimum integer r for which there exists a set of (not necessarily normalized nor orthogonal) vectors

v 1 , . . . , v r ∈ H 1 such that t = r i=1 v ⊗N i . (4)
The symmetric border rank sbr(t) of a symmetric tensor t ∈ S N H 1 is defined as the minimum integer s for which there exists a sequence of symmetric tensors t i ∈ S N H 1 of symmetric rank s tending to t, lim i→∞ t i = t. Obviously, one has

sbr(t) ≤ sr(t). (5)
From a physical standpoint, the symmetric rank sr has a simple interpretation in terms of the decomposition of a bosonic manyparticle state as the sum of Hartree states where all N atoms share the same single-particle state v i . The symmetric rank sr of a state |ψ is then the minimum number of terms that have to appear in the sum in order to exactly reconstruct the state |ψ . This decomposition can be straightforwardly reformulated within second quantization in terms of the associated polynomial. Given a homogeneous polynomial p(x 1 , . . . , x n ) of degree N in n variables, the symmetric rank sr(p) is the minimum integer r for which there exists a decomposition as the sum of r N -th powers of linear forms,

p(x 1 , . . . , x n ) = r i=1   n j=1 φ i,j x j   N (6) with φ i,j ∈ C.
Identifying again the variables x 1 , . . . , x n of the polynomial with the creation operators â † i of the atom in the i = 1, . . . , n single-particle state, the above definition of symmetric rank in terms of Hartree states is immediately recovered: it is the minimum integer r for which

|ψ = r i=1   n j=1 φ i,j â † j   N |vac . (7)
The complex coefficients φ i,j of the i-th linear form on the j-th variable of the polynomial have the physical interpretation of the j-state component of the i-th Hartree wavefunction.

Analogously the symmetric border rank sbr(p) of a homogeneous polynomial p(x 1 , . . . , x n ) of degree N in n variables is defined as the minimum integer s for which there exists a sequence of homogeneous polynomials

p i (x 1 , . . . , x n ) of degree N in n variables of symmetric rank s tending to p(x 1 , . . . , x n ), lim i→∞ p i (x 1 , . . . , x n ) = p(x 1 , . . . , x n ).
These definitions can be generalized to not necessarily symmetric tensors, e.g. the ones describing the state of distinguishable quantum particles. For a given tensor t ∈ H ⊗N 1 , its rank r(t) is defined as the minimum integer r for which there exists a family of vectors v (j) i ∈ H 1 with j = 1, . . . , N and i = 1, . . . , r, such that

t = r i=1 v (1) i ⊗ • • • ⊗ v (N ) i . ( 8
)
The border rank br(t) of a tensor t ∈ H ⊗N 1 is defined as the minimum integer s for which there exists a sequence of ranks tensors t i ∈ H ⊗N 1 such that lim i→∞ t i = t. Note that in the absence of the symmetry requirement, the single-particle wavefunctions v

(1) i , . . . , v (N ) i of the different N particles are no longer necessarily equal. The so-called Comon's conjecture [34] states that if t ∈ S N H 1 , then sr(t) = r(t). Such a conjecture is proved to be true only in few cases [34,?]. An analogous conjecture can be formulated in terms of the border rank and the symmetric border rank (see e.g. [35]).

Analogous concepts can be introduced for skew-symmetric tensors t ∈ N H 1 . In this case, the skew-symmetric rank ar(t) is defined as the minimum integer r for which there exists a family of orbitals v (j) i ∈ H 1 with j = 1, . . . , N and i = 1, . . . , r, such that

t = r i=1 v (1) i ∧ • • • ∧ v (N ) i . ( 9
)
The skew-symmetric border rank abr(t) of a skew-symmetric tensor t ∈ N H 1 is defined 1 as the minimum integer s for which there exists a sequence of skew-symmetric rank s skewsymmetric tensors t i ∈

N H 1 such that lim i→∞ t i = t. In physical terms, the decomposition (9) corresponds to expanding the many-particle state of a Fermi system as the sum of Hartree-Fock states,

|ψ = r i=1   N k=1   n j=1 φ (k) i,j f † j     |vac , ( 10
)
1 It is interesting to note that the rank of a general skew-symmetric t ∈ 2 H1 is equal to half the rank of the skew-symmetric matrix T that is obtained by writing t in an arbitrary basis of H1 [START_REF] Harris | Algebraic Geometry[END_REF].

where, φ (k) i,j is the j-state component of the k-th orbital k = 1, . . . , N ) of the i-th Hartree-Fock state (i = 1, . . . , r) and f † j are the (fermionic) creation operators for a particle in the j single-particle state; as usual for Fermi systems, creation operators satisfy the anti-commutation rules of Grassmann variables,

f † i f † j -f † j f † i = 0 (11)

Algorithm

Let now H 1 be a 2-dimensional vector space defined over the complex numbers. The algorithm to compute the symmetric rank of a homogeneous polynomial p in two variables (i.e. the symmetric rank of a symmetric tensor t ∈ S N H 1 ) is classically attributed to J. J. Sylvester. A more modern proof of it is presented in [24]. We actually refer to a simplified version of the same available in [25]. From now on we will interchangeable use the polynomial and symmetric tensor language. Before going into the details of the algorithm we need to define the so-called Catalecticant matrices [START_REF] Geramita | Commutative Algebra and Algebraic Geometry[END_REF]38] C N -r,r (p) (C N -r,r (t)) associated to a given bivariate polynomial p(x, y) (to a given tensor t ∈ S N H 1 ),

p(x, y) = N i=0 N i a i x i y N -i . (12)
The r-th Catalecticant matrix C N -r,r (p) = (c k,j ) associated to p is the (N -r+1)×(r+1) matrix with entries: c k,j = a k+j-2 with k = 1, . . . , N -r + 1 and j = 1, . . . , r + 1.

It is classically known [38] that, for a given homogeneous bivariate polynomial p(x, y) of degree N , if r is the minimum integer for which all the minors of C N -r,r (p) of order (r + 1) vanish then sbr(p) = r. Moreover in [24] and in [25] it is shown that if sbr(p) = r, then sr(p) can only be equal either to r or to N -r + 1. The way to distinguish among these last two possibilities is to verify if a certain polynomial of degree r constructed via elements in the kernel of C N -r,r (p) (see precisely below in the algorithm) has or not distinct roots. So the Algorithm that we are going to describe works as follows: for a given p(x, y) find firstly its symmetric border rank, then compute its symmetric rank, and, finally, give the decomposition in the form (6).

It is worth to remark here that, over the complex numbers, the symmetric border rank of any homogeneous p(x, y) is always bound from above by the symmetric rank of a generic bivariate polynomial, that is,

sbr(p) ≤ N + 1 2 . ( 13
)
Moreover, since if sbr(p) = r then sr(p) ∈ {r, N -r + 2}, we easily deduce that if p(x, y) is such that its symmetric border rank sbr(p) is strictly less than both its symmetric rank sr(p) and the generic symmetric rank N +1 2 , then the symmetric rank of p(x, y) is strictly bigger then the generic value

N +1 2
. We are now ready to state the Algorithm that for a given homogeneous bivariate polynomial of degree N , calculates its symmetric border rank, its symmetric rank and its decomposition as the sum of r N -th powers of linear forms as in (6).

Algorithm

Input: The projective class of a degree N homogeneous polynomial p(x, y) in two variables.

Output 1: sbr(p). Output 2: sr(p). Output 3: a decomposition of p as in (6).

1. Initialize r = 0; 2. Increment r ← r + 1; 3. Compute C N -r,r (p)'s (r + 1) × (r + 1)-minors;

if they are not all equal to zero then go to step 2 (note that this can only happen if r ≤ N +1 2 ); else, Output 1: sbr(p) = r. Go to step 4. 4. Choose a solution (u 0 , . . . , u r ) of the system

C N -r,r (p) • (u 0 , . . . , u r ) t = 0. ( 14
)
If the polynomial

q(t 0 , t 1 ) = u 0 t r 0 + u 1 t r-1 0 t 1 + • • • + u r t r 1 (15)
has distinct roots,

q(t 0 , t 1 ) = µ r k=1 (β k t 0 -α k t 1 ), (16)
then Output 2: sr(p) = r and go to step 6, otherwise Output 2: sr(p) = N -r + 2 and go to step 5. 5. Choose a solution (u 0 , . . . , u sr(p) ) of the system C N -sr(t),sr(t) (p) • (u 0 , . . . , u sr(p) ) t = 0.

The polynomial

q(t 0 , t 1 ) = u 0 t sr(p) 0 + u 1 t sr(p)-1 0 t 1 + • • • + u sr(p) t sr(p) 1 (18) has distinct roots, q(t 0 , t 1 ) = µ sr(p) k=1 (β k t 0 -α k t 1 ). (19)
Go to step 6. 6. Solve the linear system in the variables λ 1 , . . . , λ sr(p) given by p(x, y) =

sr(p) k=1 λ k (α k x + β k y) N . (20)
For such λ i 's, the decomposition (20) is our Output 3.

Observe that, for the polynomials p(x, y) for which sbr(p) < sr(p), the polynomial q(t 0 , t 1 ) defined in (15) has not distinct roots, i.e. there exist integers s 1 , . . . , s r with r < sr(p) and r k=1 s k = sbr(p) < sr(p) such that the polynomial q(t 0 , t 1 ) obtained as in step 4 of our algorithm can be written as follows,

q(t 0 , t 1 ) = µ r k=1 ( βk x 0 -αk x 1 ) s k (21)
We like to stress here that those roots have an algebraic meaning even if they are not distinct (see [25]). In fact, in this case, there exist λ1 , . . . , λr ∈ C and polynomials in two variables F 1 , . . . , F r of degrees (s 1 -1), . . . , (s r -1) respectively such that

p(x, y) = r k=1 λk (αx + βy) N -(s k -1) F k . ( 22
)
Before moving to the physical applications, we wish to stress that in the present section we have chosen to restrict our attention to the algorithm that compute the symmetric rank for a polynomial in two variables because this is the relevant case for the applications that we are going to discuss in the next section. A more general algorithm that computes the symmetric rank of a homogeneous polynomials in n ≥ 2 variables can be found in [26].

Examples of application

We now proceed to illustrate the application of the general concepts discussed in the previous sections to specific examples of current interest in the field of atomic physics. In particular, we focus our attention on the n = 2 case where each of the N bosonic atoms has only two available states, denoted as ↑ and ↓ respectively: in this case, the symmetric border rank, the symmetric rank and the decomposition can be obtained using the algorithm reviewed in Sec.3.2.

Spin Squeezed States

In the last years, spin squeezed states (SSS) have attracted quite some attention as a tool to improve the sensitivity of interferometric device for precision measurements, as they allow to overcome the so-called standard quantum limit imposed by shot noise [27,28]. Experimental demonstrations using different physical systems have been reported in [30][31][32].

To generate a spin squeezed state, one can start from a coherent spin state (CSS), i.e. a symmetric rank-1 Hartree state where all atoms share the same wavefunction. In the present case where the single atom Hilbert space is bidimensional, Hartree states can be parametrized in terms of a macroscopic spin vector moving on the surface of the so-called Bloch sphere. Its direction is specified by two angles θ, ϕ with 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π,

|coh : θ, ϕ = 1 √ N ! cos(θ/2) â † ↑ + e iϕ sin(θ/2) â † ↓ N |vac : (23)
the θ = 0 (θ = π) case with the macroscopic spin pointing along (opposite to) the quantization axis z corresponds to all atoms being in the ↑ (↓) state. A macroscopic spin along the "horizontal" plane orthogonal to the quantization axis corresponds to equally populated ↑ and ↓ states; the phase ϕ corresponds to the angle made by the macroscopic spin and the x axis.

As proposed in [28], SSS can be obtained from a θ = π/2, ϕ = 0 CSS by means of a one-axis twisting of this state via the self-phase modulation Hamiltonian,

|SSS : µ = exp(-iµS 2 z /2) |coh : θ = π/2, ϕ = 0 . ( 24
)
In elementary quantum mechanics, the total spin operator S z is defined as the sum of the corresponding S (i) z operators acting on the i-th atom and leaving the other atoms unperturbed; the ↑ and ↓ states of the i-th atom are eigenstates of S (i) z with eigenvalue ±1/2. In a second quantization formalism, the total spin operator S z has the simple form

S z = 1 2 â † ↑ â↑ -â † ↓ â↓ . ( 25
)
The first step in view of computing the rank of the |SSS state is to write it in a second quantization form as a polynomial in the creation operators â † ↑,↓ . Expanding the coherent spin state ( 23) and applying the one-axis twist operators as in (24), one easily obtains the form

|SSS : = 1 √ 2 N N ! × × N k=0 N k e -iµ(k-N/2) 2 /2 (â ↑ ) N -k (â ↓ ) k |vac . ( 26
)
The key element of the algorithm discussed in Sec.3.2 are the catalecticant matrices C N -r,r (SSS) associated to the polynomial: in our case, their entries for 1 ≤ k ≤ N -r + 1 and 1 ≤ j ≤ r + 1 are equal to

c k,j = 1 √ 2 N N ! exp -i µ 2 k + j -2 - N 2 2 . (27)
To determine whether the SSS has symmetric border rank r one has to test the r + 1-minors of the r-th catalecticant matrix C N -r,r (SSS). Let 0 ≤ j 1 , . . . , j r+1 ≤ (N -r + 1) be the row indices of the rows that are involved in a given minor. Straightforward algebraic manipulation shows that the minor is zero if and only if the Vandermonde determinant [START_REF] Lang | Linear algebra[END_REF]

1 e -iµj1 • • • e -iµj1r 1 e -iµj2 • • • e -iµj2r . . . . . . . . . . . . 1 e -iµjr+1 • • • e -iµjr+1r = 0 (28)
vanishes, i.e. if and only if

1≤jα<j β ≤(r+1) e -iµj β -e -iµjα = 0. (29)
In order for this equality to hold, one has to have µ(j β -j α ) ≡ 0 (mod 2π) for at least one pair j α = j β .

As discussed in Sec.3.2, the state has symmetric border rank equal to r if and only if the rank of the r-th catalecticant matrix C N -r,r (SSS) is not maximal (i.e. smaller than r + 1) and the rank of all the previous r -th catalecticant matrices C N -r ,r (SSS) with r < r was maximal (i.e. equal to r + 1). Assuming that this latter condition is satisfied, the symmetric border rank is r if for all possible choices for j 1 , . . . , j r+1 , there exists at least one pair j α = j β such that µ(j β -j α ) ≡ 0 (mod 2π).

(

) 30
If µ is an irrational multiple of 2π, this condition can never be satisfied, so the symmetric border rank of the spin squeezed state has the generic value g N = N +1 2 that corresponds to have maximal symmetric border rank.

If µ is a rational multiple of 2π, i.e. µ = 2πp/q (p, q are coprime integers) and q ≤ r, the pigeon hole principle guarantees that for any choice of row indices 0 ≤ j 1 , . . . , j r+1 ≤ (N -r + 1), there always exists a pair j α = j β such that µ(j α -j β ) ≡ 0 (mod 2π). On the other hand, for r < q there exists a set j 1 = 1, . . . , j r+1 = r + 1 such that all pairs µ(j α -j β ) ≡ 0 (mod 2π). Combining these two arguments into the algorithm of Sec.3.2, we conclude that the symmetric border rank of the spin squeezed state with squeezing parameter µ = 2πp/q is equal to q if q ≤ g N and equal to g N if q ≥ g N . In order to determine the value of the symmetric rank, we have to consider the polynomial q(t 0 , t 1 ) defined in (15) that is associated to vectors in the null space of C N -r,r (SSS) for r = sbr(SSS)

For µ = 2πp/q and q ≤ g N , direct inspection shows that the (r + 1)-component (r = q) vector 1, 0, . . . , 0, -e iπpq e -iπpN t (31) always belongs to the null space of the r-th catalecticant matrix C N -r,r (SSS), so that the r roots of the associated polynomial q(t 0 , t 1 ) = t r 0 -e iπpq e -iπpN t r 1 (32) are always distinct. This shows that the symmetric rank is always equal to the symmetric border rank:

sr(SSS) = sbr(SSS). ( 33
)
The situation is more delicate in the general case when µ is an irrational multiple of 2π or µ = 2πp/q with q > g N , so that the symmetric border rank has the maximal, generic value sbr = g N . If N is even, the only possibility for the symmetric rank is to be sr = sbr = g N . If N is odd, one has to assess whether sr = g N (the generic case) or sr = g N + 1.

To this purpose, one has to repeat the procedure of calculation the roots of the polynomial associated with the null space of the r = g N + 1 catalecticant matrix. By means of a MAT-LAB/OCTAVE code implementing the Algorithm described in Sec.3.2, we have checked that the roots are always distinct, so that sr = sbr = g N . These results on dependence of the symmetric rank of SSS on the squeezing parameter µ are summarized in Fig. 1.

Let us now move to the decomposition. For µ = 2π/q, the roots of the polynomial (32) are complex numbers of magnitude 1 and equidistant phases by 2π/q: this remarkable fact recovers a result originally found in [33] that the spin squeezed state with µ = 2πp/q can be written as a Schrödinger-catlike linear superposition of q macroscopically different coherent spin states of the form (23) with θ = π/2 and equidistant phases φ = φ 0 + 2πj/q, j = 0, . . . , q -1; the initial phase is φ 0 = 0 unless both p and q + N are odd, in which case is φ 0 = π/q.

The direction of the macroscopic spin is therefore always oriented along the plane orthogonal to the quantization axis and makes equidistant angles φ with the x direction. A numerical solution of the linear system at the step 6 of the algorithm presented in Sec.3.2 shows that the weigths λ i of the q coherent spin states have equal magnitude. These facts are summarized by the remarkable writing of the spin squeezed state in the form

|SSS : µ = 2πp/q = = q j=1 e iζj √ q |coh : θ = π/2, ϕ = φ 0 + 2πj/q (34)
with suitably chosen phases ζ j '2 [33].

Fock States

In the previous subsection we have seen that the symmetric rank and symmetric border rank are equal for spin squeezed states for all values of µ. Here we illustrate a class of physically relevant states for which the symmetric rank and symmetric border rank are different. Having this distinction clear in mind is essential if the concept of Schmidt rank is to be used as a measure of entanglement [22].

In the mathematical literature, it is a well known fact that the symmetric rank and the symmetric border rank of bivariate monomials usually have different values: straightforward inspection of the form of the catalecticant matrices and of the associated polynomial (15) shows that the symmetric rank of a monomial of the form x N -k y k , with k ≤ N/2, is equal to sbr = k + 1, while the symmetric rank is equal to sr = N -k + 1 ≥ sbr.

The physical states corresponding to monomials are Fock states (FS)

|ψ = (â ↑ ) †(N -k) (â ↓ ) †k |vac , (35)
where exactly k particles sit in the ↓ state and the remaining N -k particles sit in the ↑ state. The results on the symmetric rank and the symmetric border rank of monomials quoted above imply that a Fock state can be decomposed either as the exact sum of sr = N -k + 1 CSS states or as the limit of the sum of a smaller (equal only if k = N/2 and N even) number sbr = k + 1 ≤ sr of CSS states. The decomposition as an exact sum of sr CSS states has the form

(â ↑ ) †N -k (â ↓ ) †k |vac = 1 (N -k + 1) N k × × N -k j=0 â † ↑ e -i 2πkj (N -k+1)k + â † ↓ e i 2πj N -K+1 (1-k N ) N |vac , (36)
where the relative phases of the ↑, ↓ amplitudes in the sr CSS states have been carefully chosen in a way to have a complete destructive interference on all monomials âN-k

↑ â †k ↓ with k = k.
The decomposition as the → 0 limit of a sum of sbr CSS states has a similar structure:

(â † ↑ ) N -k (â † ↓ ) k |vac = 1 k n k × × lim →0 1 k   k j=0 â † ↑ + e i 2π k j â † ↓ N -k â †N ↑   |vac . (37)
In contrast to [START_REF] Harris | Algebraic Geometry[END_REF], only the monomials â †k ↓ with k < k vanish by destructive interference, while the ones with k > k disappear when the → 0 limit is taken.

Conclusions and perspectives

In this paper we have reported an example of application of algebraic geometry tools to a simple model of quantum physics. We have calculated the symmetric rank and the decomposition of spin squeezed states and Fock states of an assembly of N indistinguishable two-level bosonic atoms: the rank of spin squeezed states shows interesting features when the squeezing parameter has a rational value (in suitable units) so that the state can be decomposed as the superposition of a few macroscopically different coherent spin states. Fock states have the peculiar property that their symmetric rank and symmetric border rank have different values.

A most challenging future perspective is to generalize the decomposition algorithms to the case of a large number N of atoms and a large number m of available states and then combine them with a numerical simulation of the time-evolution of an interacting system. First attempts in this direction used the semiclassical representations of a quantum field [START_REF] Gardiner | Quantum Noise[END_REF] or a reformulation of the N -body problem in terms of stochastic Hartree [START_REF] Carusotto | [END_REF] (Hartree-Fock [44]) wavefunctions for systems of bosonic (fermionic) particles, but in many practical applications to strongly correlated many body systems this method lead to numerically untractable problems.

From a mathematical point of view, the computation of the symmetric border rank of a homogeneous polynomial of degree N in n > 2 variables (i.e. N bosonic atoms in n > 2 available states) is a still open problem as the equations defining the corresponding varieties are not yet understood (see e.g. [START_REF] Landsberg | [END_REF] and references therein). Nevertheless, algorithms to compute the symmetric rank and the decomposition have been recently developed [26] using a generalized version of the catalecticant matrices, the so-called Henkel matrices. Their application to quantum physics problems is a natural next step for our research.

The situation for the skew-symmetric tensors describing systems of N fermions is somehow specular: a criterion to determine the skew-symmetric border rank of a given skewsymmetric tensor of any N was recently developed [START_REF] Landsberg | [END_REF]. On the other hand, we are not aware of any algorithm to compute the skew-symmetric rank and the actual decomposition except for the first meaningful case N = 2 where the tensor is a matrix.

Fig. 1 .

 1 Fig. 1. Numerical plot of the symmetric rank and symmetric border rank of spin squeezed states of the form (24) as a function of the squeezing parameter µ. Total number of particles N = 13.

Acknowledgments

AB was partially supported by Project Galaad of INRIA Sophia Antipolis Méditerranée (France) and Marie Curie Intra-European Fellowships for Career Development (FP7-PEOPLE-2009-IEF): "DECONSTRUCT". IC is grateful to Ch. Miniatura and P. Vignolo for the kind hospitality at INLN and acknowledges financial support from the ERC via the QGBE grant. Stimulating discussions with P. Hyllus are warmly acknowledged.