> <\body> <\hide-preamble> ||||>> |<\doc-note> The first author has been supported by an ERC-2023-ADG grant for the ODELIX project (number 101142171). The second author has been supported by the ANR-22-CE48-0008 (OCCAM) and ANR-22-CE48-0016 (NODE) grants. <\wide-tabular> ||||||| ||LOGO_ERC-FLAG_FP.png>|61.62pt|28.51pt||>>>>> ||<\author-affiliation> CNRS, École polytechnique, Institut Polytechnique de Paris Laboratoire d'informatique de l'École polytechnique (LIX, UMR 7161) 1, rue Honoré d'Estienne d'Orves Bâtiment Alan Turing, CS35003 91120 Palaiseau, France |>>|||<\author-affiliation> CNRS, École polytechnique, Institut Polytechnique de Paris Laboratoire d'informatique de l'École polytechnique (LIX, UMR 7161) 1, rue Honoré d'Estienne d'Orves Bâtiment Alan Turing, CS35003 91120 Palaiseau, France >>||> Throughout this note, let > be a field of characteristic zero. Consider a zero-dimensional ideal ,\,x|]>>>. Modulo a linear change of coordinates, one may put in general position, in the sense that distinct roots of have distinct >-coordinates. The original shape lemma was proposed by Gianni and Mora in order to describe the shape of alexicographical Gröbner basis of . In the simplest case when the ideal is radical and in general position, they show that it has a lexicographical Gröbner basis of the form <\equation*> G=-g|)>,\,x-g|)>,g|)>|}> where \\|]>> is monic and ,\,g\\|]>> are of degree deg g>. More recently, a non-commutative version of the shape lemma was proposed in for zero-dimensional left ideals of the algebra >,\,D>|]>>, where =\,\,x|)>>> and>,\,D>> are the partial derivations with respect to ,\,x>. If is generated by operators with constant coefficients, then the \Pnon-\Q shape lemma directly follows from the classical shape lemma, since >,\,D>|]>\\,\,x|]>> as >-algebras. For the general case, the authors introduce the notion of Dradical ideals as a non-commutative counterpart of radical ideals. They then generalize the classical shape lemma to D-radical ideals. It turns out that only constant linear changes of coordinates are required in order to put D-radical ideals into general position. In this paper, we take the perspective of differential algebra. In that case, any differential ideal generated by linear differential polynomials is automatically radical. There is also a counterpart for the classical correspondence between radical ideals and solutions sets: in suitable polynomial exponential function spaces, left ideals of >,\,D>|]>> are in one-to-one correspondence with solutions sets; see, 4.3>. For any zero-dimensional left ideal of >,\,D>|]>>, we shall prove that we may put it in shape position modulo a linear change of derivations <\equation*> D>\D>+q|)>*D>+\+q|)>*D>,D>\D>,\,D>\D>, where ,\,q\\|]>> are polynomials. This corresponds to the change of coordinates <\equation*> =x,=x+p|)>,\,=x+p|)>, where =-q> for ,n>. Contrary to the shape lemma from, our change of derivations may have non-constant coefficients. On the other hand, no further assumptions (like the D-radicality assumption) are required for our result. We start with the presentation of our main result in the bivariate case. <\theorem> Let \\> be a field equipped with commuting derivations > and > such that =D=1> and =D=0>. Let be a left ideal in ,D|]>> such that the quotient \\,D|]>/I> is finite-dimensional over >. Then there exists a polynomial \> such that the powers of the projection of +D> in > span > as a >-vector space. <\proof> For an element \,D|]>>, let >> denote its projection in >. Let dim>>. We pick \> such that the dimension of the subspace spanned by +D|)>|\>> for > is largest possible. We denote this subspace by > and its dimension by , and will prove that . We also introduce \p*D+D>. We adjoin a countable set of variables ,z,z>,\> to > and let our derivatives act via >|)>\0>> and >|)>\z>>. We also define the differential operator \\+z*D> on,\|]>>>. Now consider the following element in the >-st exterior power of ,\|]>> as a ,\|]>>-module: <\equation> \\\>\>\\\>. By using the commutation rules between > and > and elements of ,\|]>>, we can write every >> in the standard differential operator form aj>*D*D> with j>\,\|]>>>. Substituting this representation into () and expanding by linearity, we can regard > as a polynomial in ,\,z>> with coefficients in >, of the following precise form: <\equation*> \=z>*|)>>*\*>|)>>*b,\,e>,where ,\,e>\>>.> By the assumption on the maximality of the dimension of >, this polynomial vanishes under any substitution of the form >=q>>, where > is a polynomial. But this implies that > must vanish itself, so all its coefficients ,\,e>> are zero in >. Consider the coefficient of the monomial >>. Since the variable >> appears only in the last power of >, we have <\equation*> \=z>*D+h,\,z>|)>. Hence the coefficient of >> in() is the wedge product of the -free parts ,>>,\,>> of the terms ,>,\,>> and the coefficient |\>> of >> in |\>>. Therefore, <\equation*> >\>\\\>\>=0. This implies that >\>. Since > and > commute and generate the whole ,D|]>> over >, we conclude that =>. <\remark> The proof above is reminiscent of one of the proofs of the cyclic vector in differential algebras (e.g., in ). However, we are not aware of adirect way to reduce our problem to the existence of cyclic vectors. The following corollary gives an alternative formulation of Theorem in terms of achange of variables. <\corollary> With the notation of Theorem, let =\>, where > and > are the partial derivatives with respect to and . Then there exists a polynomial > with the following property. Consider an invertible polynomial change of variables =x+p,=y> and denote the partial derivatives with respect to > and> by >> and >>, respectively. Then the projections of powers of >> generate >. <\proof> Applying Theorem, let \> be the resulting polynomial and take q>>. Direct computation shows that >=D> and >=q D+D>, so >> has the desired . <\example> Let us illustrate Theorem on the example ,D|)>\,D|]>>. Note that the results of are not applicable to this ideal since it is not D-radical. We consider y*D+D> and compute its powers modulo : <\eqnarray*> >||*D+D+2*y*D*D+D\2*y*D*D+D,>>|>|>|+D|)>**D+D|)>\3*D*D,>>|>|>|+D|)>*3*D*D\0.>>>> Thus, modulo , we have \D-*y*D> and \D-y*D+*y*D>. The corresponding change of variables is =x-y/2,=y>, and it transforms the ideal into <\equation*> I=>,D>-D>+*y*D>|)>. We now pursue with the multivariate case, by showing how to reduce it to the bivariate case from the previous section. <\theorem> Let \\,\,x|)>> be a field equipped with pairwise commuting derivations >,\,D>> such that >|)>=\j>> for all |i,j|\>\n>. Let be a left ideal in >,\,D>|]>> such that the quotient \\>,\,D>|]>/I> is finite-dimensional over >. Then there exist univariate polynomials ,\,q\|]>> such that the projections of powers of <\equation*> D\D>+q|)>*D>+\+q|)>*D> generate > as a >-vector space. <\proof> We proceed by induction on . The base case follows from Theorem. Assume that 2>. We apply the induction hypothesis to the ideal \>,\,D>|]>> and =\|)>> and obtain polynomials ,\,q\|]>> such that the projections of powers of <\equation*> \D>+q|)>*D>+\+q|)>*D> generate <\equation*> \\>,\,D>|]>/>,\,D>|]>>|)>. The embedding >,\,D>|]>\\>,\,D>|]>> yields an embedding \>. Then> is generated by the projection of ,D>|]>>>. We apply Theorem, this time to the ideal ,D>|]>>>> with ,x=x>, and =\,\,x|)>>. This gives us a polynomial \|]>>> such that the projections of powers of <\equation*> D\+q|)>*D>=D>+q|)>*D>+\+q|)>*D> generate ,D>|]>/\,D>|]>|)>=>. This finishes the proof. Again, we may reformulate this result in terms of changes of variables. <\corollary> With the notation of Theorem, let =\,\,x|)>>, so >,\,D>> are the respective partial derivatives. Then there exist polynomials ,\,p\|]>> with the following property. Consider an invertible polynomial change of variables <\equation*> =x,=x+p|)>,\,=x+p|)> and denote the partial derivatives with respect to ,\,> by >,\,D>>, respectively. Then the projections of powers of >> generate >. <\proof> Applying Theorem , let ,\,q\> be the obtained polynomials. We take \-q>> for ,n>. Direct computation shows that <\equation*> D>=D>+q|)>*D>+\+q|)>*D> and >= D>> for ,n>, so >> has the desired property. It is instructive to further examine the structure of modules that are generated by powers of a single derivation. For this, we recall: <\proposition> Let =\,\,x|)>> and let >,\,D>> be the partial derivatives with respect to ,\,x>, respectively. Let be a left ideal in >,\,D>|]>> such that the quotient\>,\,D>|]>>/I> is finite-dimensional over >. Then, for a differential operator \*D>+\+\*D>>, the following conditions are equivalent: <\enumerate-roman> The projections of ,\> generate > as a >-vector space. For every ,n>, contains an element of the form >-p>, where \\>. <\proof> > > >. As any element in , the projection of >> in > can be written as a >-linear combination of the projections of powers of . We can take this combination of powers as our polynomial >. > > >. Since > is spanned by power products of >>'s and each of the >>'s is congruent modulo to a polynomial in >, the projection of this polynomial ring is the whole>. This section is dedicated to a more abstract version of Theorem. We start with a lemma that is essentially contained in the proof of , but that we will reprove here for convenience of the reader. <\lemma> Let > be a field of characteristic zero equipped with pairwise commuting derivations ,\,D>, which are linearly independent over >. Let ,\,D|)>|||>>> be the column vector with entries ,\,D>>. Then there exist elements ,\,x\\> and a nonsingular matrix \n>> such that the derivations ,\,> with ,\,|)>>\A,\,D|)>>> pairwise commute and |)>=\j>> for every |i,j|\>\n>. <\proof> Linear independence of ,\,D> implies that there exist elements ,\,x> such that the vectors |)>,\,D|)>|)>> for ,n> are linearly independent. Consider amatrix whose rows are these vectors. We set B> and ,\,|)>>\A,\,D|)>>>. For every i\n>, since |)>,\,D|)>|)>>> is the -th column of , we have |)>=\>. In order to show that ,\,> pairwise commute, consider ,|]>> for j>. This Lie bracket can be written as a linear combination *+\+a*> with ,\,a\\>. On the other hand, ,|]>|)>=0> for every k\n>, so =\=a=0>. This concludes the commutativity proof. <\proposition> Let > be a field of characteristic zero equipped with pairwise commuting derivations ,\,D>, which are linearly independent over >. Let be a left ideal in ,\,D|]>> such that the quotient =\,\,D|]>/I> has finite dimension over >. Then there exist pairwise commuting >-linear combinations ,\,> of ,\,D>, which are linearly independent over>, and such that the projections of powers of > generate > as a >-vector space. <\proof> By Lemma , there exist ,\,x> and a matrix \ n>> such that ,\,|)>>\A,\,D|)>>> commute pairwise and |)>=\j>>. We apply Theorem with >=>> for ,n> and obtain a >-linear transformation of ,\,> to ,\,>, where =+q|)>*> and => for ,n>, such that the projections of powers of> span > as a >-vector space. Composing the invertible transformations ,\,D|)>\,\,|)>> and ,\,|)>\,\,|)>>, we obtain the desired change of derivations. <\bibliography|bib|tm-plain|> <\bib-list|6> B.Dwork, G.GerottoF.J.Sullivan. -functions>. Princeton University Press, 1994. P.GianniT.Mora. Algebraic solutions of systems of polynomial equations using Gröbner bases. , 356, 247\U257. 1989. J.vander Hoeven. Generalized power series solutions to linear partial differential equations. , 42(8):771\U791, 2007. M.Kauers, C.KoutschanT.Verron. A shape lemma for ideals of differential operators. , 677(1):448\U459, 2025. J.F.Ritt. , 33. Amer. Math. Soc., 1948. M.F.Singer. Model theory of partial differential fields: From commuting to noncommuting derivations. , 135:1929\U1934, 2007. <\initial> <\collection> <\attachments> <\collection> <\associate|bib-bibliography> <\db-entry|+ydUMKXAFKd|inproceedings|GM89> <|db-entry> T. > <\db-entry|+ydUMKXAFKE|article|KKV25> <|db-entry> C. T. > <\db-entry|+1jitqFVuOJN|article|vdH:lpde> <|db-entry> > <\db-entry|+ydUMKXAFKf|book|Ritt48> <|db-entry> > <\db-entry|+ydUMKXAFKe|book|DGS94> <|db-entry> G. F. J. > -functions> <\db-entry|+ydUMKXAFKP|article|Sing07> <|db-entry> > <\references> <\collection> > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> GM89 KKV25 vdH:lpde KKV25 Ritt48 DGS94 KKV25 KKV25 Sing07 <\associate|toc> |math-font-series||font-shape||1.Introduction> |.>>>>|> |math-font-series||font-shape||2.The bivariate differential shape lemma> |.>>>>|> |math-font-series||font-shape||3.Generalizations to the multivariate case> |.>>>>|> |math-font-series||font-shape||4.Application to general partial differential fields> |.>>>>|> |math-font-series||font-shape||Bibliography> |.>>>>|>