H A Gueniche

P A Glaude

G Dayma

R Fournet

F Battin-Leclerc
email: frederique.battin-leclerc@ensic.inpl-nancy.fr

RICH METHANE PREMIXED LAMINAR FLAMES DOPED BY LIGHT UNSATURATED HYDROCARBONS PART I: ALLENE AND PROPYNE

Keywords: Premixed laminar flame, methane, allene, propyne, modeling

The structure of three laminar premixed rich flames have been investigated: a pure methane flame and two methane flames doped by allene and propyne, respectively. The gases of the three flames contain 20.9% (molar) of methane and 33.4 % of oxygen corresponding to an equivalent ratio of 1.25 for the pure methane flame. In both doped flames, 2.49 % of C 3 H 4 was added, corresponding to a ratio C 3 H 4 / CH 4 of 12 % and an equivalent ratio of 1.55. The three flames have been stabilized on a burner at a pressure of 6.7 kPa using argon as dilutant, with a gas velocity at the burner of 36 cm/s at 333 K. The concentration profiles of stable species were measured by gas chromatography after sampling with a quartz microprobe. Quantified species included carbon monoxide and dioxide, methane, oxygen, hydrogen, ethane, ethylene, acetylene, propyne, allene, propene, propane, 1,2-butadiene, 1,3-butadiene, 1-butene, iso-butene, 1-butyne, vinylacetylene and benzene. The temperature was measured using a thermocouple in PtRh (6%)-PtRh (30%) settled inside the enclosure and ranged from 700K close to the burner up to 1850K.

In order to model these new results, some improvements have been made to a mechanism previously developed in our laboratory for the reactions of C 3 -C 4 unsaturated hydrocarbons. The main reaction pathways of consumption of allene and propyne and of formation of C 6 aromatic species have been derived from flow rate analyses.

INTRODUCTION

Soots and polyaromatic hydrocarbons (PAH), which are present in the exhaust gas of diesel engine, represent a high part of the urban pollution. Many efforts have then been focused on reducing the emissions of these compounds. The formation of soot precursors and PAH in combustion involves small unsaturated hydrocarbons, the chemistry of which is still very uncertain. Different reaction pathways have been proposed for the formation and the oxidation of the first aromatic compounds, involving the reactions of C 2 (acetylene), C 3 or C 4 unsaturated species [1][2][3][4][5].

As the determinant role of propargyl radicals in forming benzene, the first aromatic ring, is now well accepted, it is important to better understand their reactions. With that purpose, the oxidation of allene (propadiene) and propyne has been already studied in several oxidation conditions: shock tubes [6][7][8], flow reactor [9,10], jet-stirred reactor [7,11] and premixed flames [10,11]. Previous works in flames include studies of the influence of the addition of allene on H 2 /O 2 /Ar [12], C 2 H 2 /O 2 /Ar [13] and C 2 H 4 /O 2 /Ar [14] mixtures.

The purpose of the present paper is to experimentally investigate the structures of two premixed laminar methane flames containing propadiene and propyne, respectively, and to compare them with that of a pure methane flame containing the same mole fractions of methane and oxygen. The use of a methane flame will allow us to have a reactive mixture rich in methyl radicals and to be more representative of combustion mixtures containing larger hydrocarbons than hydrogen or C 2 flames. These results have been used to improve the mechanism previously developed in our laboratory for the reactions of C 3 -C 4 unsaturated hydrocarbons (propyne, allene, 1-butyne, 2-butyne, 1,3-butadiene) [8,15].

EXPERIMENTAL PROCEDURE

The experiments were performed using an apparatus newly developed in our laboratory to study temperature and stable species profiles in a laminar premixed flat flame at low pressure.

The body of the flat flame matrix burner, provided by McKenna Products, was made of stainless steel, with an outer diameter of 120 mm and a height of 60 mm (without gas/water connectors).

This burner was built with a bronze disk (95% copper, 5% tin). The porous plate (60 mm diameter) for flame stabilization was water cooled (water temperature: 333 K) with a cooling coil sintered into the plate. The burner could be operated with an annular co-flow of argon to favor the stabilization of the flame. This horizontal burner was housed in a water-cooled vacuum chamber evacuated by two primary pumps and maintained at 6.7 kPa by a regulation valve. This chamber was equipped of four quartz windows for an optical access, a microprobe for samples taking and a thermocouple for temperature measurements. The burner could be vertically translated, while the housing and its equipments were kept fixed. A sighting telescope measured the position of the burner relative to the probe or the thermocouple with an accuracy of 0.01 mm. The flame was lighted on using an electrical discharge. Gas flow rates were regulated by RDM 280 Alphagaz and Bronkhorst (El-Flow) mass flow regulators. The C 3 reactants (purity discussed in the text) and methane (99.95 % pure) were supplied by Alphagaz -L'Air Liquide. Oxygen (99.5% pure) and argon (99.995% pure) were supplied by Messer.

Temperature profiles were obtained using a PtRh (6%)-PtRh (30%) type B thermocouple (diameter 100 µm). The thermocouple wire was sustained by a fork and crosses the flame horizontally to avoid conduction heat losses. The junction was located at the centre of the burner.

The thermocouple was coated with an inert layer of BeO-Y 2 O 3 to prevent catalytic effects [16].

The ceramic layer was obtained by damping in a hot solution of Y 2 (CO 3 ) 3 (93% mass.) and BeO (7% mass.) followed by a drying in a Mecker burner flame. This process was reiterated (about ten times) until the whole metal was covered. Radiative heat losses are corrected using the electric compensation method [17].

The sampling probe was in silica with a hole of about 50 μm diameter (d i ). The probe was finished by a small cone with an angle to the vertical of about 20°. For temperature measurements in the flames perturbed by the probe, the distance between the junction of the thermocouple and the end of the probe was taken equal to two times d i , i.e. to about 100 μm. Gas samples were collected in a pyrex loop and directly obtained by connecting through a heated line the quartz probe to a volume, which was previously evacuated by a turbo molecular pump down to 10 -7 kPa and which was then filled up to a pressure of 1.3 kPa (pressures were measured by a MKS 0-100 Torr pressure transducer). The pressure drop between the flame and the inlet of the probe ensured reactions to be frozen.

(C 4 H 4 ), butadienes (1,2-C 4 H 6 , 1,3-C 4 H 6 ), 1-butyne (1-C 4 H 6) , 1-butene (1-C 4 H 8 ), iso-butene (i-C 4 H 8 )
and benzene (C 6 H 6 )) were analysed on a Haysep packed column by FID and nitrogen as gas carrier gas. The identification of these compounds was performed using GC/MS and by comparison of retention times when injecting the product alone in gas phase. Figure 1 what is observed due to the scale of the figure, the separation between the peaks of iso-butene, 1-butene and 1,3-butadiene and those of vinylacetylene and 1-butyne is acceptable. The peaks of propane and cyclopropane cannot be distinguished. Water and small oxygenated compounds, such as acetone, acetaldehyde and formaldehyde, were detected by GC-MS but not quantitatively analysed. Toluene could be quantified for flames of richer mixtures, but was not detected in the flames studied here due to our limit of detection (around 1x10 -6 in mole fraction).

FIGURE 1

Calibrations were performed by analysing a range of samples containing known pressures of each pure compound to quantify and mole fractions were derived from the known total pressure in the sampling line. Error on mole fractions was around + 10% for major compounds, but can be more for minor ones.

EXPERIMENTAL RESULTS

Three flames stabilized on the burner at 6.7 kPa with a gas flow rate of 3.29 l/min corresponding to a gas velocity at the burner of 36 cm/s at 333 K and mixtures containing 20.9% methane and 33.4% oxygen have been investigated: ♦ A pure methane flame supplied with a mixture containing also 45.6% argon and corresponding to an equivalent ratio of 1.25.

♦ Two doped flames supplied with mixtures containing also 43.2% argon and 2.49 % of allene or propyne and corresponding to an equivalent ratio of 1.55. C 3 compounds represented then 12% of the amount of methane. Chromatographic analysis showed that allene was 98.4% pure, containing propyne (1.2%), propene (0.25%) cyclopropane (0.08%), 1-butene (0.04%), 1,3-butadiene (0.04%), 1,2-butadiene (0.01%) and 1-butyne (0.04%) as major impurities. Propyne was 98.3% pure, but contained also allene (0.44%), propene (0.04%) cyclopropane (0.05%), 1-butene (0.6%), 1,3-butadiene (0.2%), 1,2-butadiene (0.3%) and 1-butyne (0.05%) as major impurities. The presence of cyclopropane as an impurity was mentioned by the provider.

Figure 2a presents the temperature profiles measured for the three flames without the probe.

The results are similar for both doped flames and different from those in the unseeded flame due to the difference in equivalence ratios. The lowest temperatures measured the closest to the burner are around 700 K. In the pure methane flame, the highest temperatures are reached between 0.4 and 0.6 cm above the burner and are around 1850K. For the doped flames, temperature peaks between 0.75 and 0.95 cm above the burner at around 1880K and decreases thereafter because of heat losses.

Figure 2b displays the experimental profiles obtained with and without the probe in the case of the flame doped with allene and shows that the presence of the probe induces a thermal perturbation, as it reduces the temperature of more than 100K in some places.

FIGURE 2

Figures 3 and4 present the profiles of the main species involved in the combustion of methane vs. the height above the burner for the three flames. Figure 3 displays the profiles of methane (fig. 3a) and oxygen (fig. 3b) and shows that the consumption of these species occurs further in the flames containing of allene and propyne. A displacement of the position of the maximum concentrations is also observed for carbon monoxide and C 2 compounds as shown in figures 3c and 4. The profile of carbon dioxide (fig. 3d) shows a marked inflexion point in the doped flames, which cannot be observed in the pure methane flame. Ethane (fig. 4d) is produced promptly in the three flames and reaches its maximum concentration close to the burner, around 0.2 cm in the pure methane flame and around 0.3 cm in the doped flames. The profile of ethylene (fig. 4c) peaks around 0.3 cm in the pure methane flame and around 0.4 cm in the doped flames, that of acetylene (fig. 4b) around 0.35 cm in the pure methane flame and around 0.5 cm in the doped flames and that of carbon monoxide (fig. 3c) around 0.45 cm in the pure methane flame and around 0.6 cm in the doped flames. While the maximum concentration of acetylene is strongly increased by the addition of allene and still more by that of propyne, the maximum concentrations reached for carbon oxides (fig. 3c and3d) hydrogen (fig. 4a) and ethylene are changed by a more limited factor and that of ethane is almost unmodified.

FIGURES 3 AND 4

Figure 5 presents the profiles of the observed C 3 products and shows that very small amounts of these products are also observed in the pure methane flame. The consumptions of allene and propyne (fig. 5a), as reactant, are the same in both doped flames. The most important C 3 products are allene in the propyne flame and propyne in the allene flame (fig. 5b). An important formation of propene (fig. 5c) is also encountered. The maximum concentrations of these C 3 products are located at around 0.2 cm from the burner. The fact that larger amounts of propyne and propene are observed in the allene flame in comparison with allene and propene in the propyne flame, as displayed in figures 5b and 5c, is partially due to the presence of larger concentrations of propyne (1.2%) and propene (0.25%) as an impurity in allene than it is the case for allene (0.4%) and propene (0.04%) in propyne. As it is observed in similar amounts in the three flames, we have considered that the product shown on figure 5d, which has the same retention time as propane and cyclopropane, is mainly propane.

FIGURE 5

Figure 5 also displays the profiles of benzene (fig. 5e), which present a maximum at around 0.25 cm from the burner. It shows that a larger maximum concentration of benzene is obtained in the propyne flame than in the allene flame (around a factor 2.5); this difference between both C 3 hydrocarbons, as that concerning acetylene, is in agreement with previous experimental results obtained in a jet stirred reactor operating between 800 and 1200 K, at 10 atm with stoichiometric mixtures [11].

Figure 6 presents the profiles of C 4 products and shows that 1-butene (fig. 6c), iso-butene (fig. 6e) and 1-butyne (fig. 6d) are formed close to the burner with maximum concentrations at around 0.2 cm, while 1,3-butadiene (fig. 6a), 1,2-butadiene (fig. 6b) and vinylacetylene (fig. 6f) are formed further in the flame. While the formations of 1-butyne and vinylacetylene are similar in both doped flames and that of iso-butene is larger in the allene flame, it is difficult to conclude for 1-butene, 1,3-butadiene and 1,2-butadiene, which, as stated previously, are present as an impurity in more important amounts in propyne than in allene.

FIGURE 6

Benzene and C4 compounds were not detected in the pure methane flame.

DESCRIPTION OF THE PROPOSED MECHANISM

This mechanism is an improvement of our previous mechanism that was built to model the oxidation of C 3 -C 4 unsaturated hydrocarbons [8,15] to better take into account the reactions of allene, propyne and propargyl radicals. [8] This C 3 -C 4 reaction base was built from a review of the recent literature and is an extension of our previous C 0 -C 2 reaction base [18]. This C 0 -C 2 reaction base includes all the unimolecular or bimolecular reactions involving radicals or molecules including carbon, hydrogen and oxygen atoms and containing less than three carbon atoms. The kinetic data used in this base were taken from the literature and are mainly those proposed by Baulch et al. [19] and Tsang et al. [20]. The C 0 -C 2 reaction base was first presented in the paper of Barbé et al. [18] and has been up-dated [8].

Reaction base for the oxidation of C 3 -C 4 unsaturated hydrocarbons

The C 3 -C 4 reaction base includes reactions involving C 3 H 2 (CH≡CCH••), C 3 H 3 (CH≡CCH 2 • ↔ •CH=C=CH 2 ), C 3 H 4 (allene and propyne), C 3 H 5 (3 isomers (aC 3 H 5 : •CH 2 CH=CH 2 , sC 3 H 5 : CH 3 CH=CH•, tC 3 H 5 : CH 3 C•=CH 2 )), C 3 H 6 , C 4 H 2 , C 4 H 3 (2 isomers (nC 4 H 3 : •CH=CHC≡CH, iC 4 H 3 : CH 2 =C•C≡CH ↔ CH 2 =C=C=CH•)), C 4 H 4 , C 4 H 5 (5 isomers (nC 4 H 5 : •CH=CHCH=CH 2 , iC 4 H 5 : CH 2 =CHC•=CH 2 ↔ •CH 2 CH=C=CH 2 , C 4 H 5 -1s: CH 3 CH•C≡CH, C 4 H 5 -1p: CH 2 •CH 2 C≡CH, C 4 H 5 -2: CH 3 •C≡CCH 2 )), C 4 H 6 (1,3-butadiene,
1,2-butadiene, methyl-cyclopropene, 1-butyne and 2-butyne), as well as the formation of benzene. Pressure-dependent rate constants follow the formalism proposed by Troe [21] and efficiency coefficients have been included, when the related data were available. This reaction base was built in order to model experimental results obtained in a jet-stirred reactor for methane and ethane [18], profiles in laminar flames of methane, acetylene and 1,3-butadiene [8] and shock tube auto-ignition delay times for acetylene, propyne, allene, 1,3-butadiene [8], 1-butyne and 2-butyne [15]. Thermochemical data are estimated by the software THERGAS developed in our laboratory [22], which is based on the additivity methods proposed by Benson [START_REF] Benson | Thermochemical kinetics[END_REF].

Reactions related to allene, propyne, propargyl radicals and related C 4 species

The part of mechanism, described below and given in Table I, is included in a file which also contains the two reactions bases described above and which can be used to run simulations using CHEMKIN [START_REF] Kee | Sandia Laboratories Report[END_REF]. In order to correctly model the consumption of benzene, our recent primary and secondary mechanisms for the oxidation of this species [START_REF] Costa | [END_REF] should also been added. A short mechanism of propane, 1-butene and iso-butene has also been considered based on our work on alkanes and alkenes [26], as well as four reactions of cyclopropane (isomerisation to propene, Habstraction to give allyl radical). The whole mechanism, which is available on request, includes 812 reactions involving 115 species.

TABLE I

For both propyne and allene, the pressure dependence of the rate constants of the additions of H-atoms to give C 3 H 5 radicals (reactions 6, 7, 27, 28 in Table 1) has been taken into account by considering high and low pressure limits. We have reevaluted the rate constants of the bimolecular initiations with oxygen molecule based on the work of Ingham et al. [39]. For allene (reaction 14), A was taken equal to 4 x 7.10 12 cm 3 .mol -1 s -1 , as there are four abstractable vinylic hydrogen atoms, while for propyne (reaction 33), A was taken equal to 3 x 7.10 11 cm 3 .mol -1 s -1 , as there are three abstractable allylic hydrogen atoms; for both compounds, the activation energy was set equal to the enthalpy of reaction.

The main addition to the reactions of allene was to consider the reactions of abstraction of a hydrogen atom by H atoms and OH, CH 3 , C 2 H, C 2 H 3 , C 2 H 5 and a-C 3 H 5 radicals (reactions [15][16][17][18][19][20][21][22]. These reactions, involving the abstraction of vinylic H-atoms, had been neglected in our previous work in shock tube conditions [8], but were found important here to correctly reproduce the profile of allene in the flame doped with this compound. For the rate constants, which were not available in the literature, we have used the same value as for the abstraction of hydrogen atoms from 1,3-butadiene to give the resonance stabilized iC 4 H 5 radicals [8].

Concerning allene, the isomerization to propyne (reaction 1) has been written reversible and the bimolecular initiation between two allene molecules to give allyl and propargyl radicals (reaction 3) has been taken into account as proposed by Dagaut et al [28]. We have also considered the additions of methyl radicals to the double bonds of allene to give 1-buten-2-yl (reaction 12) and iso-butenyl (reaction 13) radicals. According to Aleksandrov et al [30], the products of the addition of oxygen atoms to allene (reaction 9) are hydrogen atoms and C 2 H 3 CO radicals which decomposed rapidly to give carbon monoxide and vinyl radicals, instead of ethylene and carbon monoxide, as previously written [8].

The unimolecular initiation of propyne to give C 2 H and CH 3 radicals (reaction 25) has been revisited using the software KINGAS [40], because the activation energy proposed by Wu and Kern (100 kcal/mol [32]) was too weak compared to the enthalpy of reaction (125 kcal/mol). To improve our modeling of the formation of acetylene, we have considered the addition of H-atoms to propyne to give acetylene and methyl radicals (reaction 26) in a single step with the rate constant proposed by Hidaka et al. [27]. A second channel giving acroleine and H-atoms has been considered for the addition of OH radicals to propyne (reaction 31). For the abstraction of hydrogen atoms from propyne by oxygen atoms (reaction 39), we have used the rate constant calculated by Adusei et al. [35].

Concerning propargyl radicals, we have added the decomposition to give H-atoms and C 3 H 2 (reaction 42) and the combination with HO 2 radicals (reaction 50). In the case of the reactions leading to aromatic compounds, we have kept, the previous value of 1.10 12 cm 3 .mol -1 s -1 for the recombination of two propargyl radicals to give phenyl radicals and H-atoms (reaction 51), as it is in good agreement with what has been recently proposed by Miller et al. [42] and Rasmussen et al. [43]. We have removed from our previous mechanism the addition of propargyl radicals to allene leading to benzene and H-atoms and the recombination between allene and propargyl radicals to produce phenyl radicals and two H-atoms, which were not considered by Rasmussen et al. [43] and which induce an overprediction of benzene in the case of the flame doped with allene.

COMPARISON BETWEEN EXPERIMENTAL AND SIMULATED RESULTS

Simulations were performed using PREMIX from CHEMKIN II [START_REF] Kee | Sandia Laboratories Report[END_REF]. The presence of hydrocarbon impurities in the C 3 reactants was taken into account in simulations; i.e. the reactants in our simulations files have the C 3 and C 4 hydrocarbons composition mentioned in page 6. To compensate the thermal perturbations induced by the presence of the quartz probe and the thermocouple in the reactive mixture, the temperature profile used in calculations is an average between the experimental profiles measured with and without the quartz probe, shifted 0.1 cm away from the burner surface, as shown in figure 2b in the case of the flame doped with allene.

Figures 3 and4 show that the model reproduces satisfactorily the consumption of reactants and the formation of the main products related to the consumption of methane in the case of the three flames. The slower consumption of methane and oxygen when added a C 3 compound is well captured, as well as the similar reactivity in the two doped flames. Calculations predict well the increase in the formation of acetylene when added allene and the still higher one when added propyne. To decouple the effect due to the increase of equivalence ratio (Φ) and that induced by the presence of a C 3 unsaturated compound, these figures display also the results of a simulation performed for a flame containing 20.9% methane and 27.6% oxygen (with no C 3 additive) for Φ= 1.55, i.e. equal to that of the doped flames. As the temperature rise is mainly influenced by Φ, we have used the same temperature profile as to model the flame containing allene. This simulation shows that the experimental differences encountered for the profiles of methane, oxygen, ethane and ethylene when added allene or propyne are only due to the increase of Φ (the profiles can almost not be distinguished from those obtained in the flame doped with allene). The profiles of carbon oxides and hydrogen are also modified by the change of Φ, but they are different in the doped and pure methane flames at Φ=1.55 due to a difference in the C/O and C/H ratios. For Φ= 1.55, the profile of carbon dioxide in a pure methane flame presents also a marked inflexion points, showing that this effect is mainly related to equivalence ratio, i.e. in a rich mixture, carbon dioxide exhibits a more marked secondary behavior due to the fact that the formation of carbon monoxide is favored. In the case of acetylene, the increase between pure methane and doped flames is a factor 4 for allene and 5 for propyne, while the difference of Φ would only explain a factor around 3. There is then a specific way of formation of this species involving our C 3 additives.

Figure 5 shows that the consumption of the C 3 reactants in the doped flames is well predicted and that the peak of the profile of the concentration of the isomer of the reactant, propene and propane in both doped flames are captured within a factor better than 2. Simulations show that the formation of cyclopropane is negligible.

The simulated profiles of C 4 compounds are displayed in Figure 6. The prediction is better than a factor 3 for butadienes, 1-butyne and 1-butene in both seeded flames, which is acceptable taking into account the errors in the analyses of these minor species and the uncertainties in the rate constants used in our model. Despite that butadienes are present as an impurity in the C 3 reactant, simulations clearly show a formation of these compounds in the seeded flames.

Simulations overpredict the formation of vinylacetylene by a factor 5 in both doped flames, but reproduce well that the same amount is obtained in both doped flames. The production of iso-butene, which is a very minor product, is also badly reproduced in both doped flames.

The model reproduces well the formation of benzene in the two doped flames and the differences observed between both C 3 additives, especially.

DISCUSSION

Figure 7 displays the flows of consumption of the C 3 reactant in the two doped flames at a temperature about 720 K corresponding to a 35% conversion. The two main pathways of consumption for both compounds are additions and H-abstractions to give propargyl radicals, H-abstractions accounting for 20% of the consumption of allene and for 40% of that of propyne.

A faster production of propargyl radicals in the case of propyne, which is due to the lower activation energies of H-abstractions, can be slightly seen on the simulated profiles of these radicals displayed in fig. 8a, even if this effect is compensated by a more rapid consumption of these radicals. At low temperature, in the zone close to the burner, resonance stabilized propargyl radicals are mainly consumed by combinations with H atoms to give back the initial reactant or its isomer, with HO 2 radicals to give formaldehyde and C 2 H and OH radicals, with methyl radicals to produce 1-butyne or 1,2-butadiene or with themselves to form H atoms and phenyl radicals. At higher temperature, further in the flame, the combinations with H-atoms and HO 2 radicals become less favoured and propargyl radicals are less consumed.

Figure 8b displays the profiles of hydrogen atoms in the two doped flame and indicates that the pool of small radicals is slightly lower in the allene flame than in the propyne flame. This effect is partially due to the fact that addition of small radicals are favored in the case of allene which includes two double bonds. That explains why the integrated amount of propargyl radicals is larger in the allene flame, in which they are less consumed by combinations, than in the propyne flame, as shown in fig. 8a. This difference in the consumption of propargyl radicals explains partly why larger amounts of benzene are obtained in the allene flame as it can be seen in fig. 5e. Another consequence of the prompter formation and consumption of propargyl radicals in the propyne flame is that the formation of phenyl radicals occurs in a mixture rich in oxygen and then more favorable to the oxidation of aromatic radicals (the only reactions under the conditions of fig. 7) than to their reaction with H-atoms to give benzene.

Under the conditions of fig. 7, the branching ratio of the combination of H atoms with propargyl radicals is more than 90 % towards the formation of propyne, that explains why the formation of propyne in the allene flame is larger than that of allene in the propyne flame as shown in fig. 5b. Other reactions of propargyl radicals include reaction with oxygen molecules to give ketene and HCO radicals, and H-abstraction by H atoms and OH radicals to give C 3 H 2 , which react also with oxygen molecules. 1,3-butadiene is obtained from 1,2-butadiene. Vinyl acetylene is a degradation product from 1-butyne and 1,2-butadiene and uncertainties in the kinetics of the reactions consuming these species could explain a deteriorated prediction.

FIGURES 7 AND 8

The main additions to allene or propyne are that of OH radicals and that of H atoms. The additions of OH radicals lead to oxygenated products and account for 26 % of the consumption of allene and to 13% of that of propyne. The additions of H atoms to allene lead to t-C 3 H 5 radicals (40% of the consumption), which react rapidly with oxygen molecules to give carbon monoxide, formaldehyde and methyl radicals, and to allyl radicals (14% of the consumption).

The resonance stabilized allyl radicals react mainly by combination with H atoms to produce propene, with HO 2 radicals to give acroleine, H atoms and OH radicals and with methyl radicals to produce 1-butene. Taking into account that the rate constants concerning allyl radicals are amongst the best established in our model [37], it is surprising that the formation of propene is not better estimated by our model in the allene flame. In the propyne flame, 30 % of propene is obtained by reaction of propyl radicals with oxygen molecules. Propyl radicals derive from propane, which is produced by recombination of methyl and ethyl radicals. The addition of H atoms to propyne leads to methyl radical and acetylene and explains why the maximum of the profile of this last species is higher in the propyne flame than in the allene flame. In the allene flame, acetylene is 50% obtained from the addition of H atoms to propyne, which is formed by combination of H atoms and propargyl radicals, and 50% obtained from reactions of C 2 H (deriving from propargyl radicals) and vinyl radicals (deriving from acroleine). Very minor channels involve the addition of methyl radicals to allene to form iso-butyl and 2-butyl radicals, which lead to iso-butene and 1-butene, respectively. Cyclopropane is consumed to give allyl radicals, but there is no production channel, even if the isomerisation from propene has been considered in our mechanism.

Figure 9 displays the reaction flows of phenyl radicals in the flame seeded with allene at the position of the maximum of the benzene profile, corresponding to a temperature about 1290 K and a 75% conversion of allene. Under these conditions, phenyl radicals are almost exclusively formed by the combination of propargyl radicals (the H abstractions from benzene are still slow) and only 4% of them are consumed by combination with H atoms to form benzene. The main consumption of phenyl radicals are due to reactions with oxygen molecules to give O atoms and phenoxy radicals or H atoms and benzoquinone, which decomposes to form cyclopentadienone and carbon monoxide. Resonance stabilized phenoxy radicals react by combination with H atoms to give phenol or by decomposition to give carbon monoxide and cyclopentadienyl radicals and then cyclopentadiene, which was certainly present in too low amount to be detected in our C 3 -C 6 analysis.

FIGURE 9

CONCLUSION

This paper presents new experimental results for rich premixed laminar flames of methane seeded or not with allene and propyne, as well as some improvements made to the mechanism previously developed in our laboratory for the reactions of C 3 -C 4 unsaturated hydrocarbons.

Profiles of temperature have been measured and profiles of stable species have been obtained for 17 products, including benzene and six C 4 unsaturated compounds that are soot precursors. The use of methane as the background and consequently of a flame rich in methyl radicals favors the formation of C 4 compounds from the C 3 compounds.

The presence of C 3 additives strongly promotes the formation of acetylene, which can be mainly attributed to the addition of hydrogen atoms to propyne. The presence of allene and propyne is also responsible for the formation of benzene, which cannot be detected in the pure methane flame. This formation is totally due to the recombination of propargyl radicals and it is worth noting that simulation using a value of the rate constant close to that now admitted in the literature leads to a good agreement.

Many similarities exist between allene and propyne flames in both reactivity and products formation. Nevertheless, the pool of small radicals is slightly larger in the propyne flame, which leads to a faster consumption of propargyl radicals and a lower formation of benzene than in the flame seeded by allene.

In line with this work, studies of flames doped with 1,3-butadiene and cyclopentene are in progress using the same methodology.

TABLE 1: REACTIONS OF ALLENE, PROPYNE, PROPARGYL AND DERIVED C 4 SPECIES

The rate constants are given (k=A T n exp(-E a /RT)) in cc, mol, s, kcal units. Reference numbers are given in brackets when they appear for the first time. The reactions in bold have been added or involve a modified rate constant compared to our last mechanism [8]. ______________________________________________________________________________

Reactions of aC3H4 (CH2=C=CH2, allene)

+ CH 3

  presents a typical chromatogram of C 3 -C 6 compounds obtained for the flame doped with allene. Contrary to

Figure 1 :

 1 Figure 1: Typical chromatogram of C 3 -C 6 compounds obtained for the flame doped with allene

Figure 2 :

 2 Figure 2: Temperature profiles in the three flames: (a) experimental measurements performed

Figure 3 :

 3 Figure 3: Profiles of the mole fractions of oxygen and C 1 species in the three flames. Points are

Figure 4 :

 4 Figure 4: Profiles of the mole fractions of hydrogen and C 2 species in the three flames. Points are

Figure 5 :

 5 Figure 5: Profiles of the mole fractions C 3 species and benzene in the three flames. Points are

Figure 6 :

 6 Figure 6: Profiles of the mole fractions C 4 species in the two doped flames. Points are

Figure 7 :

 7 Figure 7: Flow rate analysis for the consumption of the C 3 reactant in (a) the flame seeded with

Figure 8 :

 8 Figure 8: Simulated profiles of (a) propargyl radicals and (b) hydrogen atoms in the doped

Figure 9 :

 9 Figure 9: Flow rate analysis for the formation and consumption of phenyl radicals in the flame

  Figure 1

  Figure 5

  Figure 9

  Stable species profiles were determined by gas chromatography. Chromatographs with a Carbosphere packed column and helium or argon as carrier gas were used to analyse O 2 , H 2 , CO and CO 2 by thermal conductivity detection and CH 4 , C 2 H 2 , C 2 H 4 , C 2 H 6 by flame ionisation detection (FID). Heavier hydrocarbons (allene (a-C 3 H 4 ),

	propyne (p-C 3 H 4 ), cyclopropane (c-C 3 H 6 ), propene (C 3 H 6 ), propane (C 3 H 8 ), vinylacetylene

Reactions of pC3H4 (CH≡ CCH3, propyne)

	FIGURE CAPTIONS
	Reactions	A	n	E a	References	No
	______________________________________________________________________________
	aC3H4 = pC3H4	2.5x10 12	0.0	59.0	Hidaka 89[27]	(1)
	aC3H4+M = C3H3+H+M	2.0x10 18	0.0	80.0	Hidaka 89	(2)
	aC3H4+aC3H4 = C3H5+C3H3	5.0x10 14	0.0	64.8	Dagaut92[28]	(3)
	C2H4+CH = aC3H4+H	1.3x10 14	0.0	-0.3	Baulch 94[20]	(4)
	C2H3+ 3 CH2 = aC3H4+H	3.0x10 13	0.0	0.0	Miller 92[3]	(5)
	aC3H4+H = aC3H5 (high pressure limit)	4.0x10 12	0.0	2.7	Wagner 72[29]	(6)
	(low pressure limit)	5.6x10 33	-5.0	4.44	Marinov96[4]
	aC3H4+H = tC3H5 (high pressure limit) 8.5x10 13	0.0	2.0	Wagner 72	(7)
	(low pressure limit)	1.1x10 34	-5.0	4.44	Marinov96
	iC4H3+ 3 CH2 = aC3H4+C2H	2.0x10 13	0.0	0.0	Miller 92	(8)
	aC3H4+O => H+C2H3+CO	6.6x10 12	0.0	3.0	Aleksandrov80[30]	(9)
	aC3H4+OH = CH2CO+CH3	2.0x10 12	0.0	-0.2	Liu 88 [31]	(10)
	aC3H4+OH = HCHO+C2H3	2.0x10 12	0.0	-0.2	Liu 88	(11)
	aC3H4+CH3 = C4H7-2	8.0x10 12	0.0	7.4	Estimated a	(12)
	aC3H4+ CH3 = iC4H7	4.0 x10 12	0.0	5.0	Estimated b	(13)
	aC3H4+O2 = C3H3+HO2	2.8x10 13	0.0	39.0	Estimated c	(14)
	aC3H4+H = C3H3+H2	1.3x10 6	2.53	9.2	Estimated d	(15)
	aC3H4+O = C3H3+OH	6.2x10 12	0.0	1.9	Estimated e	(16)
	aC3H4+OH = C3H3+H2O	6.2x10 6	2.0	0.4	Estimated d	(17)
	aC3H4+CH3 = C3H3+CH4	2.0x10 12	0.0	7.7	Wu87[32]	(18)
	aC3H4+C2H = C3H3+C2H2	1.0x10 13	0.0	0.0	Wu87	(19)
	aC3H4+C2H5 = C3H3+C2H6	5.0x10 14	0.0	19.8	Estimated d	(20)
	aC3H4+C2H3 = C3H3+C2H4	5.0x10 14	0.0	19.8	Estimated d	(21)
	aC3H4+C3H5 = C3H3+C3H6	2.0x10 12	0.0	7.7	Dagaut 92	(22)
	pC3H4+M = C3H3+H+M	4.7x10 18	0.0	80.0	Hidaka 89	(23)
	C2H2+ 3 CH2 = pC3H4	3.5x10 12	0.0	0.0	Tsang 86[20]	(24)
	pC3H4 = C2H+CH3	4.2x10 15	0.0	125.0 Estimated f	(25)
	pC3H4+H = C2H2+CH3	1.3x10 5	2.5	1.0	Hidaka 89	(26)
	pC3H4+H = tC3H5 (high pressure limit)	8.5E12	0.0	1.7	Wagner 72	(27)
	(low pressure limit)	5.6x10 25	-7.27	6.58	Marinov96
	pC3H4+H = sC3H5 (high pressure limit)	5.8E12	0.0	3.1	Wagner 72	(28)
	(low pressure limit)	3.8x10 25	-7.27	7.98	Estimated g
	pC3H4+O = CHCO+CH3	1.5x10 13	0.0	2.1	Warnatz 84 [33]	(29)
	pC3H4+OH = CH2CO+CH3	4.3x10 11	0.0	-0.8	Boodaghians 87 [34]	(30)
	pC3H4+OH =H+C2H3CHO	4.3 x10 11	0.0	-0.8	Estimated h	(31)
	pC3H4+HO2 = C2H4+CO+OH	6.0x10 9	0.0	8.0	Estimated i	(32)
	pC3H4+O2 = C3H3+HO2	2.1x10 12	0.0	40.8	Estimated c	(33)
	pC3H4+H = C3H3+H2	1.7x10 5	2.5	2.5	Estimated j	(34)
	pC3H4+CH3 = C3H3+CH4	2.2	3.5	5.7	Estimated j	(35)
	pC3H4+C2H = C3H3+C2H2	3.6x10 12	0.0	0.0	Estimated j	(36)
	pC3H4+C2H3 = C3H3+C2H4	2.2	3.5	4.7	Estimated j	(37)
	pC3H4+C2H5 = C3H3+C2H6	2.2	3.5	6.6	Estimated j	(38)
	pC3H4+O = OH+C3H3	3.4x10 4	2.16	4.8	Adusei96[35]	(39)
	pC3H4+OH = C3H3+H2O	3.1x10 6	2.0	-0.3	Estimated j	(40)
	pC3H4+HO2 = C3H3+H2O2	9.6x10 3	2.6	13.9	Estimated j	(41)

ACKNOWLEDGEMENTS

The authors thank J.F. Pauwels and P. Desgroux from the Laboratoire de Physicochimie des Processus de Combustion et de l'Atmosphère (PC2A-CNRS) of Villeneuve d'Asq and C. Vovelle and J.L. Delfau from the Laboratoire de Combustion et Systèmes Réactifs (LCSR-CNRS) of Orléans from their useful advices in the experimental part.

Reactions of C3H3 (CH≡ CCH2•, resonance stabilized propargyl radicals)

(51) ________________________________________________________________________________________ a : Rate constant of this addition of methyl radicals considered as twice the value proposed by Tsang [37] for propene assuming a fall-of effect of a factor 4 to take into account the experimental pressure of 6.7 kPa.

b : Rate constant of this addition of methyl radicals taken equal to the value measured by Tsang [38] at pressures between 1.5 and 5 bar assuming the same fall-of effect as for reaction (12). c : Rate constant of the bimolecular initiation with oxygen molecule calculated as proposed by Ingham et al [39] (see text). d : Rate constant estimated as that of the similar reaction in the case of 1,3-butadiene [8]. e : Rate constant for the H-abstraction calculated at 1000 K from the values proposed by Aleksandrov et al. [30] for rate constants of the global reaction and of the addition. f : Rate constant of this unimolecular initiation calculated by the modified collision theory at 1500 K using software KINGAS [40].

g : Rate constant at low pressure limit deduced from the rate constant at low pressure limit proposed by Marinov et al. [4] for reaction (27) and from the ratio between the rate constants at high pressure limit of reactions ( 27) and ( 28).

h : Rate constant of this addition taken equal to that of reaction (30) i : Rate constant estimated as that of the similar reaction in the case of acetylene [20]. j : Rate constant estimated as that of the similar reaction in the case of propene as proposed by Tsang [37]. k : Rate constant taken equal to that of the recombination of •H atoms with allyl radicals as proposed by Allara et al. [41]. l : Rate constant estimated according to the correlation proposed for allyl radicals by Heyberger et al. [26].


Fran Cois Boulier

E cient computation of regular di erential systems by change of rankings using K ahler di erentials

Keywords: di erential algebra, K ahler di erentials, FGLM

We present two algorithms to compute a regular di erential system for some ranking, given an equivalent regular di erential system for another ranking. Both make use of K ahler di erentials. One of them is a lifting for di erential algebra of the FGLM algorithm and relies on normal forms computations of di erential polynomials and of K ahler di erentials modulo di erential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated.

Introduction

Regular1 di erential systems permit to represent the radicals of di erential polynomial ideals as Gr obner bases or triangular sets permit to represent usual polynomial ideals. As for Gr obner bases, regular di erential systems depend on admissible orderings, called rankings. The Rosenfeld{Gr obner algorithm [START_REF] Cois | Etude et implantation de quelques algorithmes en alg ebre di erentielle[END_REF][START_REF] Fran Cois Boulier | Representation for the radical of a nitely generated di erential ideal[END_REF] which computes regular di erential systems given any nite system of polynomial ODE or PDE and any ranking R is implemented in the diffalg package of the MAPLE VR5 standard library.

In this paper, we are concerned by the problem: given a di erential system A = 0; S 6 = 0 regular w.r.t. some ranking R, de ning some regular di erential ideal r and given some ranking R 6 = R, compute a di erential system à = 0; S 6 = 0 regular w.r.t. R and equivalent to A = 0; S 6 = 0 i.e. de ning the same regular di erential ideal r.

A straightforward solution consists of course of running the Rosenfeld{Gr obner algorithm over A = 0; S 6 = 0 for the new ranking R but this is not e cient. The algorithms described in this paper use the fact that a representation of r by a regular di erential system is already available in order to compute à = 0; S 6 = 0.

The paper is motivated by the following fact. Let be a di erential system that we want to study by representing it by regular di erential ideals. The rankings which provide the most interesting informations (most of the time, elimination rankings) usually make the computation utterly memory and time expensive while there usually exists rankings which require nearly no computation (most of the time, orderly rankings).

The situation is indeed very similar to that of Gr obner bases (but perhaps still more striking): the Buchberger's algorithm is often very e cient for total degree orderings and often very ine cient for elimination orderings. In the particular case of a zero dimensional ideal a of a polynomial ring R = K X], the algorithm FGLM 11] solves the problem we consider: computing a Gr obner of a for an admissible ordering, knowing a Gr obner basis of a for some di erent ordering. FGLM relies on the following principles.

1. A polynomial 1 t 1 + + s t s (where the 's are coe cients in K and the t's are terms over X) lies in a if and only if the terms t 1 ; : : : ; t s are linearly dependent over K in the factor ring R=a. 2. Detecting a linear dependency between t 1 ; : : : ; t s in R=a amounts to detect a linear dependency between the normal forms of t 1 ; : : : ; t s , assimilating these latter to vectors of elements of K. 3. The known Gr obner basis permits to compute normal forms. The FGLM algorithm then enumerates the terms t 1 ; : : : ; t s by increasing order w.r.t. the new ordering. The relations found are the polynomials of the new Gr obner basis. The hypothesis dima = 0 implies there are only nitely many irreducible terms w.r.t. any Gr obner basis of a and ensures the termination of FGLM.

The algorithms we present for di erential algebra are not as e cient as FGLM (apart perhaps the algorithm for systems of linear PDE which is a special case and very close to FGLM) but solve anyway the problem under consideration. They proceed in three steps:

1. rst compute only the set of leaders v 1 ; : : : ; v t of Ã. More precisely, one computes t sets of derivatives W 1 ; : : : ; W t such that v i is the greatest derivative of W i (for each i) w.r.t. the new ranking R and such that r \ K W i ] 6 = (0). See sections 4 and 5 ; 2. knowing W i , compute a nonzero polynomial f i 2 r \ K W i ] for each i (section 6) ; 3. use f 1 ; : : : ; f t in order to speed up the computation of à = 0; S 6 = 0 using Rosenfeld{ Gr obner. The second step, determining f i knowing W i , is performed by applying exactly the same principles as FGLM. To carry it out, we had to de ne a normal form of a di erential polynomial modulo a regular di erential ideal r (algorithm NF). The known regular di erential system permits to compute normal forms.

The rst step relies on the computation of K ahler di erentials [START_REF] Johnson | K ahler di erentials and di erential algebra[END_REF][START_REF] Helaman | Analysis of PSLQ, An Integer Relation Finding Algorithm[END_REF] which \linearize the problem". To perform it, we give two algorithms (we assume for a while that r is a prime ideal of a di erential polynomial ring R and denote G the di erential eld of fractions of R=r).

The rst algorithm (called K ahler) readily applies a key theorem (theorem 5) on K ahler di erentials, using a coding trick, and calling Rosenfeld{Gr obner.

The second one (algorithm dfglm) can be viewed as a lifting of FGLM for regular di erential systems but it only applies for di erential systems the solutions of which depend on nitely many arbitrary constants. It is however more e cient than K ahler. It applies the following principles:

1. there exists a nonzero di erential polynomial in r\K w 1 ; : : : ; w s ] (where w 1 ; : : : ; w s are derivatives) if and only if the K ahler di erentials of w 1 ; : : : ; w s are linearly dependent over G in G=K (theorem 4) ;

2. detecting a linear dependency over G between these di erentials amounts to detect a linear dependency between their normal forms, assimilating these latter to vectors of elements of G (theorem 8) ;

3. the known regular di erential system A = 0; S 6 = 0 permits to compute normal forms of K ahler di erentials in G=K (algorithm DNF). The dfglm algorithm then enumerates all the derivatives of the di erential indeterminates by increasing order w.r.t. the new ranking. The hypothesis that the solutions of r only depend on nitely many arbitrary constants plays the same role as the zero dimension hypothesis in FGLM and ensures the termination of dfglm.

In general the di erential ideal r is not prime but its total ring of fractions is isomorphic to a direct product of di erential elds which all admit a same transcendence basis over K (proposition 1) and we explain how to handle the general case.

A pedagogic example is completely discussed. Applying our methods over Euler's equations for a perfect uid, we prove the pressure satis es an autonomous fth order PDE (section 7). Fact which does not seem to be known.

A secondary result of the paper is the algorithm fglm for linear PDE which only applies for systems of linear PDE (section 8). It is given a linear di erential system A = 0 (there are no inequations when the system is linear) regular w.r.t. a ranking R the solutions of which depend on nitely many arbitrary constants and a new ranking R 6 = R. It does compute the desired di erential system à = 0 regular w.r.t. R (not only its set of leaders). It is very close to the original FGLM. It may be useful for solving a system of linear PDE A by seeking ODE in the di erential ideal A] and running a linear ODE solver e.g. 7]. This idea was already developed by 20] with a di erent method. We apply our algorithm over a famous example of E. Cartan.

We do not address complexity issues.

Di erential algebra

We only provide a short presentation. The reference books are 21] and 15]. We also refer to the MAPLE VR5 diffalg package and thus to the articles 4, 5] which present it. An example is provided in section 1.1.

A derivation over a ring R is a map : R ! R which satis es, for every a; b 2 R . If a i > b i for each 1 i m then = = a 1 ?b 1 1 am?bm m . A di erential ideal a of R is an ideal of R stable under derivation i.e. such that a 2 a ) a 2 a:

Let A be a nonempty subset of R. We denote (A) the ideal generated by A. We denote A] and p A] the di erential ideal and the radical of the di erential ideal generated by A which are respectively the smallest di erential ideal and the smallest radical di erential ideal which contain A. If a is an ideal of R and S = fs 1 ; : : : ; s t g, we denote a : S 1 the saturation of a by S which is the ideal a : S 1 = fp 2 R j 9a 1 ; : : : ; a t 2 N; such that s a 1 1 s at t p 2 ag: Let U = fu 1 ; : : : ; u n g be a set of di erential indeterminates. Derivation operators apply over di erential indeterminates giving derivatives u. We denote U the set of all the derivatives. Let K be a di erential eld. The di erential ring of the di erential polynomials built over the alphabet U with coe cients in K is denoted R = KfUg.

A ranking is a total ordering over the set of the derivatives 15, page 75] satisfying the following axioms 1. v > v for each derivative v and derivation , 2. v > w ) v > w for all derivatives v; w and each derivation .

One distinguishes orderly rankings, which satisfy: ord > ord ) u > v for all u; v 2 U from elimination rankings which satisfy: u > v ) u > v for all u; v 2 U and ; 2 : Fix a ranking. The greatest indeterminate v occuring in a di erential polynomial p is called the leader of p. The leading coe cient of p w.r.t. v is called the initial of p. The di erential polynomial @p=@v is called the separant of p. Assume p = 2 K. Let v be the leader of p and d = deg(p; v). A di erential polynomial q is said to be partially reduced w.r.t. a di erential polynomial p = 2 K if no proper derivative of v occurs in q. It is said to be reduced w.r.t. p if it is partially reduced w.r.t. p and deg(q; v) < d.

A set A of di erential polynomials is said to be di erentially triangular if it is triangular and if its elements are pairwise partially reduced. It is said to be autoreduced if its elements are pairwise reduced.

If A is a set of di erential polynomials and v is a derivative then A v = f p j ld p vg.

Thus R v denotes the set of all the di erential polynomials having leader less than or equal to v.

A pair fp 1 ; p 2 g of di erential polynomials is said to be a critical pair if the leaders of p 1 and p 2 are derivatives of some same di erential indeterminate u (say ld p 1 = 1 u and ld p 2 = be formed with any two elements of A. Let fp 1 ; p 2 g be a critical pair. Denote 12 the least common multiple between 1 and 2 and assume 12 where s 1 ; s 2 denote the separants of p 1 and p 2 . Let A = 0; S 6 = 0 be a system of di erential polynomial equations and inequations. The critical pair fp 1 ; p 2 g is said to be solved by A = 0; S 6 = 0 if there exists a derivative v < 12 u such that (p 1 ; p 2 ) 2 (A v ) : (S \ R v ) 1 : De nition 1 (regular di erential systems) A di erential system A = 0; S 6 = 0 of a di erential polynomial ring R is said to be a regular di erential system (for a ranking R) if C1 A is di erentially triangular, C2 S contains the separants of the elements of A and is partially reduced w.r.t. A, C3 all the pairs fp; p 0 g 2 pairs(A) are solved by A = 0; S 6 = 0 (coherence property).

If A = 0; S 6 = 0 is a regular di erential system then the ideal A] : S 1 (resp. (A) : S 1 ) is called the regular di erential ideal (resp. regular algebraic ideal) de ned by the system.

The Rosenfeld{Gr obner algorithm 3, 4, 5] is implemented in the MAPLE VR5 diffalg package. Given any nite family of di erential polynomials and any ranking, it represents the radical of the di erential ideal ] generated by as an intersection of regular di erential ideals presented by regular di erential systems. p ] = A 1 ] : S 1 1 \ \ A t ] : S 1 t :

If A = 0; S 6 = 0 is a regular di erential system, we call derivatives under the stairs of A the elements of U which are not derivatives of any leader of element of A. See section 1.1 for an explanation of this terminology. Denote N this set and L the set of the leaders of the elements of A. Then K L; N] is the ring of the di erential polynomials partially reduced w.r.t. A.

Regular systems enjoy the following properties. See 5].

Theorem 1 Let A = 0; S 6 = 0 be a regular di erential system of R = KfUg. Let L denote the set of leaders of A and N the set of the derivatives under the stairs of A. Then the regular algebraic ideal (A) : S 1 is radical (Lazard's lemma) ; if b denotes a prime ideal minimal over (A) : S 1 then the set N furnishes a transcendence basis of the eld of fractions of R=b over K (Lazard's lemma) ;

we have A] : S 1 \ K L; N] = (A) : S 1 (Rosenfeld's lemma) ;

the regular di erential ideal A] : S 1 is radical (lifting of Lazard's lemma) ; there is a bijection between the prime di erential ideals p 1 ; : : : ; p t which are minimal over A] : S 1 and the prime ideals b 1 ; : : : ; b t which are minimal over (A) : S 1 given by p i \ K L; N] = b i (lifting of Lazard's lemma) ; the system A = 0; S 6 = 0 admits a purely algebraic solution, viewed as a polynomial system of K L; N], if and only if it admits a di erential solution ;

every purely algebraic solution of the system A = 0; S 6 = 0, viewed as a polynomial system of K L; N], can be extended in a unique way as a di erential solution.

Proposition 1 seems to be new.

Proposition 1 Let A = 0; S 6 = 0 be a regular di erential system of R and p 1 ; : : : ; p t be the di erential prime components of A] : S 1 . Let K i be the di erential eld of fractions of R=p i . Then the total ring of fractions of R= A] : S 1 is isomorphic to the direct product of di erential elds G = K 1 K t .

Proof Let 1 i 6 = j t be two indices. By the Chinese Remainder Theorem 10, Exercise 2.6, page 79] it is su cient to prove that the sum p i +p j = (1) in G. Since A]:S 1 is radical by the lifting of Lazard's lemma, this amounts to prove that there exists f i 2 p i and f j 2 p j such that f i + f j belongs to none of the p.

Let X denote the nite set of derivatives occuring in A S. Let b i = p i \ K X]. By the lifting of Lazard's lemma again b 1 ; : : : ; b t are the minimal primes of (A) : S 1 in K X]. By Lazard's lemma, all these ideals have the same dimension d. We claim b i + b j is not contained in the union of the b. On one hand, if it were, it would be contained in one of them by the prime avoidance lemma and would have dimension2 d. On another hand, b i + b j is a proper divisor of b i and b j since both are minimal over (A) : S 1 . Thus dimb i + b j < d.

Thus there exists f i 2 b i and f j 2 b j such that f i + f j belongs to none of the b. The polynomial f i + f j 2 K X] thus it belongs to none of the p by the lifting of Lazard's lemma. Since f i 2 p i and f j 2 p j , the proposition is proved.

An example

We will follow the next example throughout this paper. It is a system of three di erential polynomial equations.

< :

u 2 x ? 4u = 0; u xy v y ? u + 1 = 0; v xx ? u x = 0: There are two derivations @=@x and @=@y and two di erential indeterminates u; v (meaning we are looking for two functions v(x; y) and u(x; y) of two independent variables). We denote u x = @u=@x and u xy = @ 2 u=@x@y. v xx ? u x ; 4v y u + u x u y ? u x u y u; u 2 x ? 4u; u 2 y ? 2u and S = fs 2 ; s 3 ; s 4 g where s 2 = 4u, s 3 = 2u x and s 4 = 2u y is the set of the nonconstant separants of the elements of A. The set of leaders of the elements of A w.r.t. R is L = fv xx ; v y ; u x ; u y g. The following diagrams show the sets of derivatives of the di erential indeterminates u and v. The leaders are presented by black circles. The areas which contain their derivatives are striped. The set of derivatives lying in the nonstriped areas is the set N of the derivatives under the stairs. The set N = fv x ; v; ug is nite here but does not need in general to be so. When nite, its cardinal is an invariant of the ideal 3 and gives the number of arbitrary constants the solutions of the system depend on. Here are the solutions of , computed using the diffalg package, the arbitrary constants being denoted c 0 ; c 1 and c 2 . u(x; y) = c 0 + c 3 x + c 4 y + x 2 + 2 c 4 c 3 xy + 1 2 y 2 ; v(x; y) = c 1 + c 2 x ? c4 c 3 ? c 4 c 3 c 0

4 c 0 y + c 3 2 x 2 + c 4 xy + c 0 c 3 y 2 + 1 3 x 3 + c 4 c 3 x 2 y + 1 2 xy 2 + c 4 6 c 3 y 3 :
The other constants c 3 and c 4 are algebraic over c 0 , c 1 and c 2 . They satisfy: c 2 3 = 4 c 0 ; c 2 Finally, let's say that we would like to compute a di erential system à = 0; S 6 = 0 regular w.r.t. the following elimination ranking R and such that A] : S 1 = Ã] : S1 .

> u x > u y > u > > v xx > v xy > v yy > v x > v y > v:
Running Rosenfeld{Gr obner directly over and R makes the memory of the computer explode.

Gr obner bases

Gr obner bases are presented in 9, 2]. Let R = K X] be a polynomial ring. A term over X is a power product of elements of X. If B is a Gr obner basis then ???! B denotes the reduction by the basis B, using the classical reduction algorithm of the Gr obner basis theory, which rewrites a term as a polynomial.

Let S = fs 1 ; : : : ; s t g. To each s k we associate an indeterminate s k over R and denote S = fs 1 ; : : : ; s t g. A Gr obner basis of S ?1 (A) is obtained by computing a Gr obner basis of (A fss ? 1 j s 2 Sg). Modulo the relation ss ? 1, each s = 1=s. See 10, Exercise 2.2, page 79].

2 Normal form of a di erential polynomial Let A = 0; S 6 = 0 be a regular di erential system. Denote L the set of its leaders and N the derivatives under the stairs. Given any di erential polynomial f 2 R, the following algorithm (Ritt's algorithm of partial reduction) computes a triple h; h; r] such that h is a power product of s's and h is the corresponding product of s's and r 2 K L; N] is a di erential polynomial satisfying h f = r (mod A]); f = h r (mod S ?1 A]):

By prem(f; g; u) we denote a function which computes the pseudo{remainder of the polynomial f by the polynomial g, viewed as univariate polynomials in the indeterminate u. partial rem(f; A) begin h := 1; h := 1; r := f while r = 2 K L; N] do let w be the highest derivative w.r.t. the ranking which appears in r and is also a proper derivative of the leader v of some p 2 A let 2 be such that v = w and s denote the separant of p h := h s deg(r;w) ; h := h s deg(r;w) ; r := prem(r; p; w) od return h; h; r] end Lemma 1 Let f f 0 (mod A]:S1 ) be two polynomials and denote h; h; r] = partial rem(f; A) and h 0 ; h 0 ; r 0 ] = partial rem(f 0 ; A). Then hr h 0 r 0 (mod S ?1 (A)).

Proof Using the speci cations of the reduction algorithm we have rh 0 r 0 h (mod A]:S 1 ) whence, using Rosenfeld's lemma, rh 0 r 0 h (mod (A) : S 1 ). Multiply both sides by hh 0 , simplify by hh = h 0 h 0 = 1 and use the fact that S ?1 ((A) : S 1 ) = S ?1 (A).

Notice that, if h; h; r] = partial rem(f; A), then the fraction r=h is not necessarily a canonical representative of f. Consider again the example of section 1.1 and take f = 2v yy .

Then r=h = 4u=u x . Take f 0 = u x . Then r 0 =h 0 = u x =1. Observe we have r=h 6 = r 0 =h 0 though 2v yy u x (mod A] : S 1 ) for u 2 x ? 4u 2 A] : S 1 . NF(f; A) begin h; h; r] := partial rem(f; A) let B be a Gr obner basis of S ?1 (A) let f be such that h r ???! B f return f end Theorem 2 The polynomial NF(f; A) belongs to K L; N; S] and we have f NF(f; A) (mod S ?1 A]):

It is a canonical representative of the residue class of f in S ?1 R=S ?1 A].

Proof The rst claim comes from the speci cations of the reduction algorithm partial rem and of the Gr obner basis reduction algorithm 2, proposition 5.27]. Let's assume now that f f 0 (mod A] : S 1 ). Let h; h; r] = partial rem(f; A) and h 0 ; h 0 ; r 0 ] = partial rem(f 0 ; A). By lemma 1 we have rh r 0 h 0 (mod (B)). Since a polynomial which is irreducible by a Gr obner basis B is a canonical representative of its residue class modulo (B) 2, proposition 5.38 (vi)], the canonicity claim is proved.

Using our implementation of the NF algorithm, we nd NF(2v yy ; A) = u x = NF(u x ; A). This example shows also that theorem 3 below would not hold at all if NF did not compute canonical representatives ! This theorem is important: it basically says that determining if there exists a linear dependency between di erential polynomials p 1 ; : : : ; p t in a factor ring R= A] : S 1 amounts to determine if there exists a linear dependency between their normal forms, regarding them as vectors of elements of K. Theorem 3 Let p 1 ; : : : ; p t 2 R be di erential polynomials. There exists 1 ; : : : ; t 2 K such that 1 p 1 + + t p t 2 A] : S 1 if and only if 1 NF(p 1 ; A) + + t NF(p t ; A) = 0. Proof On one hand p 2 A] : S 1 if and only if NF(p; A) = 0 (theorem 2); on another one 3 K ahler di erentials See 10, chapter 16] for a presentation of K ahler di erentials in the purely algebraic case and 13] for K ahler di erentials in di erential algebra.

De nition 2 Let K be a eld. If G is an algebra over K then the module of K ahler di erentials of G over K, denoted G=K is the module over G generated by the set fd(b) j b 2 Gg The following theorem is purely algebraic.

Theorem 4 If K is a eld of characteristic zero and G is a eld extension of K then the elements 1 ; : : : ; r of G are algebraically independent over K if and only if d( 1 ); : : : ; d( r ) are linearly independent over G.

Proof 10, theorem 16.14, page 400] or 13, lemma, page 94]. Theorem 5 Let K be a di erential eld. If G is a nitely generated di erential eld extension of K, say G = Kh 1 ; : : : ; r i, then G=K is generated by d( 1 ); : : : ; d( r ) as a di erential vector space over G. v xx ? u x ; 4v y u + u x u y ? u x u y u; u 2 x ? 4u; u 2 y ? 2u:

We are looking for a di erential system à = 0; S 6 = 0 equivalent to A = 0; S 6 = 0 for the elimination ranking

> u x > u y > u > > v xx > v xy > v yy > v x > v y > v:
The K ahler di erentials in R=K of the elements of A are

d(A) 8 > > < > > :
d(v xx ) ? d(u x ); 4ud(v y ) + (u x ? u y u)d(u x ) + (u x ? u x u)d(u y ) + (4v y ? u x u y )d(u); 2u x d(u x ) ? 4d(u); 2u y d(u y ) ? 2d(u):

Let's assume that the di erential ideal A] : S 1 is prime and denote G the di erential eld of fractions of R= A] : S 1 . We apply theorem 5 to compute the set of leaders of Ã. To make the set d(A) generates G=K as a di erential vector space over G, we may just 1. de ne two new di erential indeterminates du and dv ; 2. code the di erentials d( u) and d( v) which occur in d(A) as derivatives du and dv of the new di erential indeterminates ; 3. enlarge the so transformed system d(A) with the equations A = 0 and the inequations4 S 6 = 0 in order to have the coe cients of the di erentials taken in G (i.e. mod A]:S 1 ) ;

8 > > > > > > > > > > < > > > > > > > > > > :
dv xx ? du x ; 4udv y + (u x ? u y u)du x + (u x ? u x u)du y + (4v y ? u x u y )du; 2u x du x ? 4du; 2u y du y ? 2du; v xx ? u x ; 4v y u + u x u y ? u x u y u; u 2 x ? 4u; u 2 y ? 2u: 4. run Rosenfeld{Gr obner over the enlarged system for the suitable ranking:

(a) the derivatives of du and dv are ranked according to R ;

(b) every derivative of du or dv is ranked higher than any derivative of u or v ;

(c) the derivatives of u and v are ranked according to R (so that Rosenfeld{Gr obner does not waste time modifying the equations of A). The Rosenfeld{Gr obner algorithm quickly computes a regular di erential system. We do not give the equations in R which are the ones of A. The other ones are du = u y u x 2u dv y ; dv xx = u y u dv y ; dv xy = u x 2u dv y ; dv yy = u y 2u dv y : We can immediately deduce from this computation that the leaders of à are u; v xx ; v xy ; v yy and, using theorem 4, that for each set W 1 = fu; v y g, W 2 = fv xx ; v y g, W 3 = fv xy ; v y g and W 4 = fv yy ; v y g we have A] : S 1 \ K W i ] 6 = (0).

The following diagram shows the derivatives of u and v. The leaders of the elements of à are presented by black circles. The areas which contain their derivatives are striped. We may then verify that the number of derivatives lying under the stairs for the ranking R is the same as for R. What if A] : S 1 is not prime ? We may consider (proposition 1 and theorem 6) that we perform the computations separately modulo the di erential prime components of A] : S 1 . We give in section 6.1 a method to verify the correctness of the result. [START_REF] Fran Cois Boulier | Representation for the radical of a nitely generated di erential ideal[END_REF] The algorithm dfglm 5.1 Normal form of a K ahler di erential Consider again the example of section 1.1. The system d(A) may be viewed as a rewrite system which rewrites the di erentials of the leaders of the elements of A as linear combinations of d(w) where w 2 N with coe cients in K L; N; S]. Recall s denotes the formal inverse of s.

d(A) 8 > > < > > : d(v xx ) ! d(u x ); d(v y ) ! ?s 2 (u x ? u y u)d(u x ) ? s 2 (u x ? u x u)d(u y ) ? s 2 (4v y ? u x u y )d(u); d(u x ) ! 4s 3 d(u); d(u y ) ! 2s 4 d(u):
We denote ???! d(A) the reduction by the rewrite system d(A). For instance, 3d(v xx ) + d(v) ???! d(A) 3d(u x ) + d(v):

Let's generalize and consider any regular di erential system A = 0; S 6 = 0 of R. Denote L the set of leaders of A and N the set of derivatives lying under the stairs. Consider the following algorithm. Proof The speci cations of the reduction algorithm imply that d is a linear combination of d(w) where w 2 L N with coe cients in K L; N; S]. Since d(A) rewrites the d(w) where w 2 L in terms of the d(w 0 ) where w 0 2 N, the rst claim is proved.

The second claim comes from the facts that if p 2 S ?1 A] then d(p) = 0 in G=K , that f = hr (mod S ?1 A]) and that A S ?1 A].

Proposition 4 Let A = 0; S 6 = 0 be a regular di erential system of R and G the total ring of fractions of R= A] : S 1 . For every 1 ; : : : ; t 2 G and w 1 ; : : : ; w t 2 N, if 1 d(w 1 ) + + t d(w t ) = 0 in G=K (1)

then 1 = = t = 0.
Proof Let p 1 ; : : : ; p n be the di erential prime components of A] : S 1 . Denote G i the eld of fractions of R=p i . By proposition 1 and theorem 6 G=K ' G 1 =K Gn=K . If a nontrivial relation (1) held in G=K then such a nontrivial relation would hold in some G i =K too and by theorem 4 the set N would be algebraically dependent modulo p i . This contradiction to theorem 1 proves the proposition. Theorem 7 Let A = 0; S 6 = 0 be a regular di erential system. Denote G = S ?1 R=S ?1 A].

The di erential DNF(

f; A) is a canonical representative of d(f) in G=K .
Proof Assume d(f) = d(f 0 ) in G=K . We have DNF(f; A) = DNF(f 0 ; A) in G=K by proposition 3. Both these di erentials are linear combinations of d(w) where w 2 N. Proposition 4 implies their coe cients are pairwise equal (as elements of G). Since these coe cients are denoted by canonical representatives (proposition 2) the di erentials DNF(f; A) and DNF(f 0 ; A) are syntactically equal. Theorem 8 Assume A] : S 1 is prime. Let fv 1 ; : : : ; v t g be a set of derivatives. Then K v 1 ; : : : ; v t ] \ A] : S 1 6 = (0) if and only if there exist 1 ; : : : ; t 2 K L; N; S] such that 1 DNF(v 1 ; A) + + t DNF(v t ; A) ???! B 0 where B is a Gr obner basis of the ideal S ?1 (A) (the reduction applying on the coe cients of the di erential).

Proof Denote G the fraction eld of R= A]:S 1 . By de nition, K v 1 ; : : : ; v t ]\ A]:S 1 6 = (0) if and only if the images in G of the derivatives v 1 ; : : : ; v t are algebraically dependent over K i.e. (theorem 4) if and only if there exists 1 ; : : : ; t 2 G such that

1 d(v 1 ) + + t d(v t ) = 0 in G=K (2)
Multiplying the coe cients by some nonzero element of G to clear the denominators, replacing them by their normal forms and substituting DNF(v i ; A) to each d(v i ) (using proposition 3) we see relation ( 2) is equivalent to

1 DNF(v 1 ; A) + + t DNF(v t ; A) = 0 in G=K (3)
where the coe cients belong to K L; N; S]. By proposition 3 the di erential on the left hand side of ( 3) is a linear combination of d(w) where w 2 N. Thus by proposition 4 relation (3) holds if and only if all its coe cients are zero modulo S ?1 A] i.e. (using Rosenfeld's lemma) if and only if they are all reduced to zero by the Gr obner basis B of S ?1 (A).

Assume A] : S 1 is prime. The above theorem permits us to look for the existence of a di erential polynomial in A]:S 1 \K v 1 ; : : : ; v t ] by interpreting DNF(v 1 ; A); : : : ; DNF(v t ; A) as vectors and performing (say) gaussian elimination. A Gr obner basis of S ?1 (A) being su cient to test equality with zero. This is applied in the dfglm algorithm of the next section. The ideal A]:S 1 is prime if and only if (A):S 1 is prime (this is a corollary to Rosenfeld's lemma). So the primality test is algorithmic.

Assume A]:S 1 is not prime. Then the total ring of fractions of R= A]:S 1 is isomorphic to a product of elds K 1 K n (proposition 1). We may run the gaussian elimination algorithm over the product and consider we are computing in parallel over each component. If a linear combination of DNF(v i ; A) is reduced to zero then it is zero over all the components; if nonzero, then it is nonzero over at least one of the components. More satisfactory, each time we need to invert some element in G, we could test if it is invertible or not (this is algorithmic by 5, corollary 4.1, point 3] but rather expensive). If it is not then a splitting of the ideal A] : S 1 is discovered and computations can go on by considering separately the two cases. This is the same idea as the one applied in commutative algebra in 16, 17, 1].

It is also sometimes possible to perform computations as if G were a eld and verify the correctness of the result afterwards (section 6.1).

An analogue of the FGLM algorithm

The following algorithm applies theorem 8 in the case of a regular di erential system A = 0; S 6 = 0 such that the set N of derivatives under the stairs is nite and the ideal A] : S 1 is prime. In that case, G=K is a nite vector space over the eld of fractions of R= A] : S 1 .

The algorithm is directly inspired from the FGLM algorithm 11]. Assume A = 0; S 6 = 0 is a regular di erential system for some ranking R. Given another ranking R, we are looking for a regular di erential system à = 0; S 6 = 0 such that A] : S 1 = Ã] : S1 . The algorithm dfglm below returns the list of the leaders of Ã.

to see is a list of derivatives to consider. This list is ordered increasingly w.r.t. R. new leaders is the list of the leaders of the elements of Ã. new irr is the list of the derivatives which are not derivatives of any element of Ã. the function call update(v; to see) inserts the derivatives of v w.r.t. all the derivations 1 ; : : : ; m in the list to see. Duplicates are removed. The list is sorted increasingly w.r.t. R. dfglm(A = 0; S 6 = 0; R) begin to see := the list of the di erential indeterminates u 1 ; : : : ; u n sorted increasingly w.r. Why should dfglm be better than K ahler ? We do not have any proof of that conjecture but a strong hint: the completion process performed by Rosenfeld{Gr obner over systems of linear PDE is close to the completion process performed by the Buchberger's algorithm (it is the same when the linear PDE depend on only one di erential indeterminate and have constant coe cients) while the behaviour of dfglm is close to the one of FGLM. And it is known that computing a Gr obner basis by change of orderings using FGLM is much faster than calling the Buchberger's algorithm.

An example

We detail the computation over the system given in section 1.1. We assume the di erential ideal A] : S1 is prime. We have s 2 = 1=(4u) and s 3 = 1=(2u x ) and s 4 = 1=(2u y ). The di erentials DNF(v; A) are linear combinations of d(u), d(v) and d(v x ). We take for R the elimination ranking > u x > u y > u > > v xx > v xy > v yy > v x > v y > v: 1. Initially to see = v; u]. The lists new irr and new leaders are empty. 2. The derivative v is picked from to see and stored in new irr. We have DNF(v; A) = d(v).

After update we get to see = v y ; v x ; u]. A] :S 1 \ K W] 6 = (0). We are looking for a nonzero polynomial. For this, we enumerate all the terms t 1 ; : : : over W by increasing total degree. At every step j we consider t 1 ; : : : ; t j and we apply theorem 3 to search 1 ; : : : ; j 2 K such that 1 t 1 + + j t j 2 A] : S 1 or to determine no such coe cients exist. Continuing the example of section 1.1 and applying this method, we obtain the system  8 > > < > > : 6.1 Verifying the correctness of the result Now, we may verify A] : S 1 = Ã] : S1 by verifying 5, corollary 4.1] on one hand that A Ã] : S1 and no element of S divides zero modulo Ã] : S1 , on another hand that à A] : S 1 and that no element of S divides zero modulo A] : S 1 . This nal veri cation proves that the computations we performed assuming A] : S 1 was prime were correct. 7 Euler's equations for an incompressible uid Written as a system of polynomial di erential equations, Euler's equations for an incompressible uid in two dimensions are (example taken from 19]) 8 < :

v 1 t + v 1 v 1 x + v 2 v 1 y + p x = 0; v 2 t + v 1 v 2 x + v 2 v 2 y + p y = 0; v 1
x + v 2 y = 0: The di erential indeterminates are v 1 ; v 2 ; p where v 1 and v 2 are the two coordinates of the speed and p is the pressure. The derivations are @=@x; @=@y and @=@t. The base eld is the eld K = Q of the rational numbers.

For some orderly ranking, the Rosenfeld{Gr obner algorithm applied over returns a unique regular di erential system A = 0 (the leaders appear on the left hand side of the equations) A 8 > > < > > :

p xx = ?2v 2 x v 1 y ? 2(v 2 y ) 2 ? p yy ; v 1 t = ?v 2 v 1 y ? p x + v 2 y v 1 ; v 1 x = ?v 2 y ; v 2 t = ?v 1 v 2
x ? v 2 v 2 y ? p y :

The system A is orthonomic (i.e. all leaders appear linearly and the initials are 1). This proves the di erential ideal ] = A] is prime 15, lemma 2, page 167]. A derivative of each di erential indeterminate appears as a leader of some element of A. The idea was: enumerating the derivatives of the pressure p by increasing order and apply theorem 8 to determine the existence of a relation in the di erential ideal. Observe we are not in the hypotheses of the dfglm algorithm for the solutions of do not depend on nitely many arbitrary constants (i.e. N is in nite). We can however use this algorithm by halting computations as soon as a leader is found. We make sure to nd one because the di erential dimension is zero. However, the coe cients of the normal forms of the K ahler di erentials were huge and made the memory of the computer explode. So we applied the fact that A is orthonomic which implies that the factor ring K L; N]=(A) is a free algebra: the sum and the product of two normal forms is still a normal form. For this reason, we could evaluate the normal forms as integer numbers to simplify computations. This is because of this use of evaluation that we say we only nearly proved our claim.

We tried afterwards to seek a di erential polynomial in ] \ K X] using theorem 3 but could not succeed (even evaluating normal forms to oating point numbers and using PSLQ 12]) because of the large number of monomials to consider.

A linear example of Cartan

The situation is simpler in the case of a system of linear PDE (there is no need of K ahler di erentials). We illustrate it over the following system of six linear PDE. ? 1 2 (x 2 ) 2 V 1

x 2 + V5 x 2 = 0; ?x 2 x 3 V 1 x 3 + x 2 V4

x 3 ? V 3 + V 4 x 1 ? x 3 V 1

x 1 + x 3 V 4

x 4 ? (x 3 ) 2 V 1

x 4 + 1 2 (x 2 ) 2 V 4

x 5 ? 1 2 (x 2 ) 2 x 3 V 1 x 5 = 0; ?(x 2 ) 2 V 1

x 3 + x 2 V 3

x 3 ? V 2 + V 3 x 1 ? x 2 V 1

x 1 + x 3 V 3

x 4 ? x 3 x 2 V 1

x 4 + 1 2 (x 2 ) 2 V 3

x 5 ? 1 2 (x 2 ) 3 V 1 x 5 = 0; ? 1 2 (x 2 ) 3 V 1 x 3 + x 2 V 5

x 3 ? V 2 x 2 + V 5

x 1 ? 1 2 (x 2 ) 2 V 1 x 1 + x 3 V 5

x 4

? 1 2 x 3 (x 2 ) 2 V 1 x 4 + 1 2 (x 2 ) 2 V 5

x The symmetries of are then given by vector elds V = (V 1 ; : : : ; V5 ) with ve components such that the Lie derivative of each element of w.r.t. V is equal to zero modulo . Computing modulo we rewrite d(x 3 ); d(x 4 ); d(x 5 ) in terms of d(x 1 ) and d(x 2 ). We thus get a system of four linear equations in d(x 1 ) and d(x 2 ) (one of them is identically zero) with linear PDE in the V i and their derivatives for coe cients. The symmetries are thus given by the common zeros of these coe cients. This is the system , computed with the determine function of liesymm.

Conclusion

We believe this paper contributes to prove that being able to compute normal forms is important since it allows to perform easily linear algebra in factor structures. In particular, the algorithms presented in this paper rely on the computation of the normal form of a fraction p=s in S ?1 R=S ?1 (A) where A = 0; S 6 = 0 is a triangular system. This computation is possible using Gr obner bases methods but there does not seem to be any known method based on triangular sets and pseudo{reduction 5 . This is a pain for we implemented since 1998 versions (joint work with Fran cois Lemaire) of Rosenfeld{Gr obner in MAPLE and C++ using di erent versions of the lextriangular triangularization algorithm 16, 17] instead of Gr obner bases and we would like to completely avoid these latter.

The author would like to thank M. Petitot for indicating him theorem 4, for the algorithm K ahler and for the analysis of Cartan's example.

  (a + b) = a + b; (a b) = ( a)b + a( b): A di erential ring is a ring endowed with nitely many derivations which commute pairwise. The commutative monoid generated by the derivations is denoted . Its elements are the derivation operators = a 1 1 am m where the a i are nonnegative integer numbers. The sum of the exponents a i , called the order of the operator , is denoted ord . The identity operator is the unique operator with order 0. The other ones are called proper. If = b

  + b 0 ) = d(b) + d(b 0 ) for all b; b 0 2 G d(b b 0 ) = b d(b 0 ) + b 0 d(b) for all b; b 0 2 G d(a) = 0 for all a 2 K: From the de nition, follows the fact that d(a=b) = (d(a) b ? a d(b))=b 2 for every a=b 2 G. Proposition 2 If K is a di erential eld and G is a di erential algebra over K then G=K has a canonical structure of di erential module over G such that d(b) = d( b) for all b 2 G and derivation over G. Proof 13, proposition, page 93].

Proof 13 , 10 4

 1310 lemma, page 94]. Theorem 6 If G 1 ; : : : ; G r are algebras over K and G = G 1 The algorithm K ahler Let's consider again the system of R = KfUg given in section 1.1.

  DNF(f; A) begin let B be a Gr obner basis of S ?1 (A) h; h; r] := partial rem(f; A) d := (d(r)h ? rd(h))h 2 let d be such that d ???! d(A) ???! B d return d end Proposition 3 Denote G = S ?1 R=S ?1 A]. The di erential DNF(f; A) is a linear combination of d(w) where w 2 N with coe cients in K L; N; S]. It is equivalent to d(f) in G=K .

  The derivatives ocuring in are u x ; u; v y ; u xy ; v xx .Let's x the following ranking R. This is the ranking w.r.t. which computations are nearly immediate.> v xx > v xy > v yy > u xx > u xy > u yy > v x > v y > u x > u y > v > u:The leaders of the elements of w.r.t. R are u x ; u xy ; v xx . Running the Rosenfeld{Gr obner algorithm over and R, we get only one regular di erential ideal

			p ] = A] : S 1
	where	A	8 > > < > > :

  3. The derivative v y is picked from to see. Its di erential Thus v y is stored in new irr. After update we get to see = v x ; v yy ; v xy ; u]. 4. The derivative v x is picked from to see. Its di erential DNF(v x ; A) = d(v x ) is not linearly dependent on DNF(v; A) and DNF(v y ; A). Thus v x is stored in new irr. After update we get to see = v yy ; v xy ; v xx ; u]. 5. The derivative v yy is picked from to see. Its di erential Thus v yy is stored in new leaders. 6. The derivative v xy is picked from to see. Its di erential DNF(v xy ; A) = 2s 4 d(u) satis es a linear relation with DNF(v y ; A). Thus v xy is stored in new leaders. 7. The derivative v xx is picked from to see. DNF(v xx ; A) = s 4 (u x u y ? 4v y )d(u) satis es a linear relation with DNF(v y ; A). Thus v xx is stored in new leaders. 8. The derivative u is picked from to see. Its di erential DNF(u; A) = d(u) satis es a linear relation with DNF(v y ; A) thus is stored in new leaders. 9. The list to see is empty. The list u; v xx ; v xy ; v yy ] of the leaders of à is returned. The dfglm algorithm can easily be transformed to provide with each new leader v a set of This algorithm follows either K ahler or dfglm. Let W be a set of derivatives such that

	6 Searching a polynomial knowing the alphabet
	DNF(v y ; A) = u x u y ? 4v y 4	d(u)
	is not linearly dependent on DNF(v; A) = d(v). DNF(v yy ; A) = s 4 (u x u y ? 4v y ) 2	d(u)
	satis es a linear relation with DNF(v y ; A). derivatives v 0 1 ; : : : ; v 0 t such that A] : S 1 \ K v; v 0

  Since the ranking is orderly, the ideal has di erential dimension zero 15, theorem 6, page 115]. This proves that ] \ Kfpg 6 = (0). Observe it is not di cult to compute a nonzero di erential polynomial in ] \ Kfv 1 g or in ] \ Kfv 2 g (see 19] for a sixth order polynomial and 3, page 94] for a fth order one). It is however a challenge to compute some nonzero di erential polynomial belonging to ] \ Kfpg ! Using theorem 8 we could solve a rst step of this problem by (nearly) proving that ] \ K X] 6 = (0) where X is the following alphabet of 39 derivatives: X = fp ttxxx ; p ttxxy ; p ttxyy ; p ttyyy ; p txxxx ; p txxxy ; p txxyy ; p txyyy ; p tyyyy ; p xxxxx ; p xxxxy ; p xxxyy ; p xxyyy ; p xyyyy ; p yyyyy ; p ttxx ; p ttyy ; p txxx ; p txxy ; p txyy ; p tyyy ; p xxxx ; p xxxy ; p xxyy ; p xyyy ; p yyyy ; p txx ; p txy ; p tyy ; p xxx ; p xxy ; p xyy ; p yyy ; p xx ; p xy ; p yy ; p x ; p y g

  5 ? 1 Ã := ; new irr := ; while to see 6 = the empty list do v := rst(to see) to see := tail(to see) if v is not a derivative of any element of new leaders then if there exists some w 2 K (w 2 new irr) s.t. NF(v; A) = P

	new leaders := new leaders fvg à := à fv ? P w wg else new irr := new irr fvg	w NF(w; A) then
	to see := update(v; to see)

od à end

We precise some of the terms used in this introduction in the next section.

u). If A is a set of di erential polynomials then pairs(A) denotes all the pairs that can

We use the fact that if p p 0 are two prime ideals then dimp dimp 0 ; if moreover dimp = dimp 0 then p = p 0 . See for instance 15, proposition 4, page 20].

= 2 c 0 ; c 0 6 = 0:[START_REF] Becker | Gr obner Bases: a computational approach to commutative algebra[END_REF] It is the degree of algebraic transcendency of the eld of fractions of R=p over K, where p is any di erential prime component of A] : S 1 .

NF(p 1 ; A) + + t NF(p t ; A) is equal to its normal form.

The inequations are important, to avoid useless splittings.

; : : : ; v 0 t ] 6 = ;. Over the above example, the algorithm would return fu; v y g; fv xx ; v y g; fv xy ; v y g; fv yy ; v y g]: This is the same answer as the one given by K ahler !

(x 2 ) 4 V 1 x

= 0; ?x 3 V 1 x 2 + V[START_REF] Cois | Etude et implantation de quelques algorithmes en alg ebre di erentielle[END_REF] x 2 = 0; ?x 2 V 1x 2 + V 3 x 2 = 0:Applying the Rosenfeld{Gr obner algorithm over for some orderly ranking R, we nd a regular di erential system A = 0 made of linear PDE with coe cients in Q(x 1 ; : : : ; x 5 ) the solutions of which depend on 14 arbitrary constants. There are no inequations since the elements of A are linear. Here is the ranking R if j j > j j then V i > V j for any i; j, if j j = j j and > w.r.t. the lex. order x 1 > > x 5 then V i > V j for any i; j, if i < j then V i > V j . We do not give A which is a bit too large, just its set of leaders:V 5 x 4 x 5 x 5 ; V 5 x 5 x 5 x 5 ; V 3 x 4 x 4 ; V 4 x 4 x 4 ; V 5 x 4 x 4 ; V 3 x 4 x 5 ; V 4 x 4 x 5 ; V 2 x 5 x 5 ; V 3 x 5 x 5 ; V 4 x 5 x 5 ; V 1 x 1 ; V 2 x 1 ; V[START_REF] Becker | Gr obner Bases: a computational approach to commutative algebra[END_REF] x 1 ; V 4 x 1 ; V 5 x 1 ; V 1 x 2 ; V 2 x 2 ; V 3 x 2 ; V 4 x 2 ; V 5 x 2 ; V 1 x 3 ; V 2 x 3 ; V 3 x 3 ; V 4 x 3 ; V 5 x 3 ; V 1 x 4 ; V 2

Running fglm for linear PDE over A = 0 and R we have got in a few seconds the desired system à = 0. Here is the set of leaders of Ã:V 1 x 1 ; V 1 x 2 ; V 1 x 3 ; V 1 x 4 ; V 1 x 5 ; V 2 ; V 3 ; V 4 x 1 x 1 ; V 4 x 1 x 4 ; V 4 x 4 x 4 ; V 4x 2 ; V 4 x 3 ; V 4 x 5 ; V 5x 3 x 3 x 3 ; V 5 x 3 x 3 x 5 ; V 5 x 3 x 5 x 5 ; V 5 x 4 x 5 x 5 ; V 5 x 5 x 5 x 5 ; V 5 x 1 x 1 ; V 5 x 1 x 2 ; V 5 x 1 x 3 ; V 5 x 1 x 4 ; V[START_REF] Fran Cois Boulier | Representation for the radical of a nitely generated di erential ideal[END_REF] x 1 x 5 ; V 5 x 2 x 2 ; V 5 x 2 x 3 ; V 5 x 2 x 4 ; V 5 x 2 x 5 ; V 5

After this paper was written, this problem was solved by Fran cois Lemaire and the author in

6].

x 4 ; V 1 x 5 : We are looking now for a di erential system à = 0 regular w.r.t. the following ranking R and such that A] = Ã]. The ranking R is the elimination ranking V 1 > > V 5 where the derivatives of each V i are ranked w.r.t. the orderly ranking: if j j > j j then V i > V i , if j j = j j then V i > V i if > for w.r.t. lexical ordering x 1 > > x 5 .

Applying the Rosenfeld{Gr obner algorithm over and R takes a lot of time. We interrupted the computation after a few minutes. Now, for linear PDE, we can design a variant of the FGLM algorithm which does compute the desired di erential system à = 0 (and not only its set of leaders !) regular w.r.t. R in a few seconds, starting from the di erential system regular w.r.t. R. Indeed, this variant of FGLM is very close to FGLM. 8.1 A note on this system

The system arises during the determination of the Lie symmetries of the dynamical system below, rst studied by E. Cartan 8]. See 18] for the mathematical theory. The system has ve state variables x i and two commands u 1 (t); u 2 (t) which are arbitrary functions.

x 1 t = u 1 ; x 2 t = u 2 ; x 3 t = u 1 x 1 ; x 4 t = u 1 x 2 ; x 5 t = 1 2 u 1 (x 2 ) 2 : We are interested in the vector elds which generate the Lie symmetries of which leave t invariant. Such symmetries are local di eomorphisms which transform the ve variables x i and map any admissible trajectory of to another one. We transform as a Pfa an system: d(x 1 ) = u 1 d(t); d(x 2 ) = u 2 d(t); d(x 3 ) = u 1 x 1 d(t); d(x 4 ) = u 1 x 2 d(t); d(x 5 ) = 1 2 u 1 (x 2 ) 2 d(t): Eliminating d(t) and the commands, we get three forms of Pfa in ve variables. d(x 3 ) = x 2 d(x 1 ); d(x 4 ) = x 3 d(x 1 ); d(x 5 ) = 1 2 (x 2 ) 2 d(x 1 ): We enlarge them with one 2{form (below), computed using the close function of the liesymm package of MAPLE. Let's call the so closed system. d(x 1 ) ^d(x 2 ) = 0: