Yet another differential shape lemma Joris van der Hoevena, Gleb Pogudinb 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 a. Email: vdhoeven@lix.polytechnique.fr b. Email: gleb.pogudin@polytechnique.edu September 18, 2025

. 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.

Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

1.Introduction

Throughout this note, let be a field of characteristic zero. Consider a zero-dimensional ideal . 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 [2] in order to describe the shape of a lexicographical 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

where is monic and are of degree .

More recently, a non-commutative version of the shape lemma was proposed in [4] for zero-dimensional left ideals of the algebra , where and are the partial derivations with respect to . If is generated by operators with constant coefficients, then the “non-commutative” shape lemma directly follows from the classical shape lemma, since as -algebras. For the general case, the authors introduce the notion of D-radical 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 are in one-to-one correspondence with solutions sets; see, e.g., [3, section 4.3].

For any zero-dimensional left ideal of , we shall prove that we may put it in shape position modulo a linear change of derivations

where are polynomials. This corresponds to the change of coordinates

where for . Contrary to the shape lemma from [4], 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.

2.The bivariate differential shape lemma

We start with the presentation of our main result in the bivariate case.

Proof. For an element , let denote its projection in . Let . We pick such that the dimension of the subspace spanned by for is largest possible. We denote this subspace by and its dimension by , and will prove that . We also introduce .

We adjoin a countable set of variables to and let our derivatives act via and . We also define the differential operator on . Now consider the following element in the -st exterior power of as a -module:

(1)

By using the commutation rules between and and elements of , we can write every in the standard differential operator form with . Substituting this representation into (1) and expanding by linearity, we can regard as a polynomial in with coefficients in , of the following precise form:

By the assumption on the maximality of the dimension of , this polynomial vanishes under any substitution of the form , where is a polynomial. But this implies that must vanish itself [5, p.35], so all its coefficients are zero in . Consider the coefficient of the monomial . Since the variable appears only in the last power of , we have

Hence the coefficient of in (1) is the wedge product of the -free parts of the terms and the coefficient of in . Therefore,

This implies that . Since and commute and generate the whole over , we conclude that .

Remark 2. The proof above is reminiscent of one of the proofs of the cyclic vector theorem in differential algebras (e.g., in [1, Theorem III.4.2]). However, we are not aware of a direct way to reduce our problem to the existence of cyclic vectors.

The following corollary gives an alternative formulation of Theorem 1 in terms of a change of variables.

Proof. Applying Theorem 1, let be the resulting polynomial and take . Direct computation shows that and , so has the desired property.

Example 4. Let us illustrate Theorem 1 on the example . Note that the results of [4] are not applicable to this ideal since it is not D-radical. We consider and compute its powers modulo :

Thus, modulo , we have and . The corresponding change of variables is , and it transforms the ideal into

3.Generalizations to the multivariate case

We now pursue with the multivariate case, by showing how to reduce it to the bivariate case from the previous section.

Theorem 5. Let be a field equipped with pairwise commuting derivations such that for all . Let be a left ideal in such that the quotient is finite-dimensional over . Then there exist univariate polynomials such that the projections of powers of

generate as a -vector space.

Proof. We proceed by induction on . The base case follows from Theorem 1. Assume that . We apply the induction hypothesis to the ideal and and obtain polynomials such that the projections of powers of

generate

The embedding yields an embedding . Then is generated by the projection of . We apply Theorem 1, this time to the ideal with , and . This gives us a polynomial such that the projections of powers of

generate . This finishes the proof.

Again, we may reformulate this result in terms of changes of variables.

Corollary 6. With the notation of Theorem 5, let , so are the respective partial derivatives. Then there exist polynomials with the following property. Consider an invertible polynomial change of variables

and denote the partial derivatives with respect to by , respectively. Then the projections of powers of generate .

Proof. Applying Theorem 5, let be the obtained polynomials. We take for . Direct computation shows that

and for , 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 [4, Theorem 8]:

Proof. i ii. As any element in A, 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 .

ii i. 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 .

4.Application to general partial differential fields

This section is dedicated to a more abstract version of Theorem 5. We start with a lemma that is essentially contained in the proof of [6, Lemma 2.2], but that we will reprove here for convenience of the reader.

Proof. Linear independence of implies that there exist elements such that the vectors for are linearly independent. Consider a matrix whose rows are these vectors. We set and . For every , since is the -th column of , we have .

In order to show that pairwise commute, consider for . This Lie bracket can be written as a linear combination with . On the other hand, for every , so . This concludes the commutativity proof.

Proof. By Lemma 8, there exist and a matrix such that commute pairwise and . We apply Theorem 5 with for and obtain a -linear transformation of to , where and for , such that the projections of powers of span as a -vector space. Composing the invertible transformations and , we obtain the desired change of derivations.

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