Complete invariant theory for multivariable dimension polynomials of finitely generated filtered modules over Ore polynomial rings

For a finitely generated left module $M$ over an Ore polynomial ring $D=K[X]$ whose indeterminates are partitioned into $p$ blocks, and for any excellent $p$-dimensional filtration of $M$, A. B. Levin (2007) proved the existence of a numerical polynomial $\phi(t_1,\ldots,t_p)$ that agrees with $\dim_K M_{r_1\ldots r_p}$ for all sufficiently large multi-indices, computed it from Gröbner bases taken with respect to the $p$ term orderings induced by the partition, and identified a first list of data of its canonical binomial expansion that do not depend on the chosen generators or filtration (total degree, leading coefficient, the maximal elements of the support under the $p!$ lexicographic orders and the coefficients at those elements, and the coefficients of all monomial terms of total degree). The open problem addressed here is the completion of this invariant theory: determine exactly which data of the canonical expansion are invariants of $M$, and determine how the invariants obtained from different partitions of the indeterminates are related. A 2008-2026 line of work by Levin, Dönch, Fürst, Landsmann, Winkler, Zhou, Huang and others has produced a steady stream of new invariants and new types of multivariate dimension polynomials without producing such a classification; the ISSAC 2020 paper in that line reports only partial invariant results (the top-degree coefficient, a simplest-form criterion, and some relationships across subsets of the derivation set).

Background

Hilbert's theorem linking the growth of a graded module to a numerical polynomial has analogues throughout algebra. In differential algebra the analogue is the differential dimension polynomial introduced by E. R. Kolchin in 1964 (announced in a 1964 Bulletin note): if $L=K\langle\eta_1,\ldots,\eta_n\rangle$ is a field extension of a differential field $K$ (a field carrying commuting derivations $\delta_1,\ldots,\delta_m$), then there is a polynomial $\omega_{\eta|K}(t)\in\mathbb{Q}[t]$ whose value at all sufficiently large integers $r$ equals the transcendence degree over $K$ of the field generated by all derivatives $\theta\eta_j$ of total order at most $r$; its degree, leading coefficient and one further coefficient are independent of the chosen system of generators of the extension, and the leading coefficient equals the differential transcendence degree (see Kolchin's monograph and the survey monograph of Kondratieva, Levin, Mikhalev and Pankratiev). Such polynomials measure the "strength" of a system of partial differential equations in the sense of Einstein - the number of free Taylor coefficients at each order of differentiation - and the same questions recur for difference (shift-operator) extensions.

A. B. Levin's 2007 paper extended this picture to several variables at once. Let $D=K[X]$ be an Ore polynomial ring: the associative algebra generated over a field $K$ by indeterminates $x_1,\ldots,x_n$ with skew multiplication $x_i a=\alpha_i(a)x_i+\delta_i(a)$ for $a\in K$, where each $\alpha_i$ is an injective endomorphism of $K$ and each $\delta_i$ is an $\alpha_i$-derivation (a skew derivation). This covers ordinary polynomial rings, rings of differential operators, and rings of difference operators. Fix a partition of the indeterminates into $p$ blocks $X_1,\ldots,X_p$ with $|X_i|=n_i$. The block-order $\operatorname{ord}_i\theta$ of a monomial $\theta$ counts the exponents of variables in $X_i$ only, and $D$ carries the $p$-dimensional filtration whose component $D_{r_1\ldots r_p}$ is spanned by the monomials with $\operatorname{ord}_i\theta\leq r_i$ for all $i$. A $p$-dimensional filtration $\{M_{r_1\ldots r_p}\}$ of a finitely generated left $D$-module $M$ is called excellent if some shift of it is stable under the ring filtration (there is a tuple $(h_1,\ldots,h_p)$ with $D_{r_1\ldots r_p}M_{h_1\ldots h_p}=M_{r_1+h_1,\ldots,r_p+h_p}$), the condition under which Hilbert-type growth statements hold. Levin proved that there is a numerical polynomial $\phi(t_1,\ldots,t_p)$ with $\phi(r_1,\ldots,r_p)=\dim_K M_{r_1\ldots r_p}$ for all sufficiently large tuples, that $\phi$ admits a canonical integer-coefficient expansion in the binomial basis $\phi(t_1,\ldots,t_p)=\sum_{0\leq i_k\leq n_k}a_{i_1\ldots i_p}\prod_{k=1}^{p}\binom{t_k+i_k}{i_k}$ with $\deg_{t_k}\phi\leq n_k$, and that $\phi$ can be computed from a Gröbner basis of the relation module of $M$ taken with respect to the $p$ term orderings $<_1,\ldots,<_p$ induced simultaneously by the partition (a multi-order Gröbner basis, constructed by a Buchberger-type algorithm whose $S$-polynomial reductions are staged across the orders), by counting the terms that survive all $p$ orders via inclusion-exclusion. In the differential-algebraic specialization, where $D$ is a ring of differential operators and the module is a module of differentials, $\phi$ becomes a multivariable differential dimension polynomial $\Phi_\eta$ of a field extension associated with a partition of the derivation set, whose invariant data are birational invariants of the extension; for modules over a Weyl algebra (the algebra of differential operators with polynomial coefficients), the one-variable specialization is the classical Bernstein polynomial introduced by I. N. Bernstein in 1971.

The central structural question about any such polynomial is which of its data are invariants of $M$: an invariant is a datum of the canonical expansion that takes the same value for the polynomials obtained from every finite system of generators of $M$ and every excellent $p$-dimensional filtration. Levin's Theorem 4.5 identifies a first list: the total degree $d=\deg\phi$, the leading coefficient $a_{n_1\ldots n_p}$, and, for each of the $p!$ lexicographic orders on $\mathbb{N}^p$, the maximal elements of the support $\Sigma_\phi=\{(i_1,\ldots,i_p)\mid a_{i_1\ldots i_p}\neq 0\}$ together with the coefficients at those maximal elements, and the coefficients $a_{i_1\ldots i_p}$ of all monomial terms of total degree $d$. In the one-variable case this collapses to Kolchin's three classical invariants. What the theorem does not provide - and what nearly two decades of subsequent work (2008-2026) has not provided - is completeness of any invariant list or a dictionary between the invariants obtained from different partitions of the indeterminates.

Problem Statement

Let $D=K[X]$ be an Ore polynomial ring over a field $K$ (each structure map $\alpha_i$ injective), let $X$ be partitioned into $p\geq 1$ blocks of sizes $n_1,\ldots,n_p$, and let $M$ be a finitely generated left $D$-module with an excellent $p$-dimensional filtration. Let $\phi_M(t_1,\ldots,t_p)$ be the associated multivariable dimension polynomial, written in its canonical binomial expansion $\phi_M(t_1,\ldots,t_p)=\sum_{0\leq i_k\leq n_k}a_{i_1\ldots i_p}\prod_{k=1}^{p}\binom{t_k+i_k}{i_k}$ with integer coefficients $a_{i_1\ldots i_p}$, so that $\deg_{t_k}\phi_M\leq n_k$ for every $k$. An invariant of $M$ (for the fixed partition) is any datum of this expansion that takes the same value for every polynomial obtained from $M$ by any other choice of finite generating set and excellent $p$-dimensional filtration. Determine the complete invariant theory of these polynomials, in the full generality of the setting above (no restrictions on the partition, the block sizes, the number of generators, or the Ore data beyond those stated):

1. Completeness of invariants. Give a necessary and sufficient characterization of the invariants of $\phi_M$: exactly which data of the canonical binomial expansion are independent of the presentation. In particular, determine whether the known list (total degree; the coefficient $a_{n_1\ldots n_p}$; for each of the $p!$ lexicographic orders on $\mathbb{N}^p$, the maximal elements of the support $\Sigma_{\phi_M}=\{(i_1,\ldots,i_p)\mid a_{i_1\ldots i_p}\neq 0\}$ and the coefficients at those elements, and the coefficients $a_{i_1\ldots i_p}$ of all monomial terms of total degree $d=\deg\phi_M$) is complete, and if it is not, extend it to a complete list, proving invariance of every listed datum and variability of every unlisted datum.

2. Relations across partitions. Determine how the invariant data of dimension polynomials associated with different partitions of $X$ are related - in particular, how the data for a finer partition are related to, or determined by, the data for the coarser partitions it refines (including the one-block partition, whose invariant data are those of the classical one-variable dimension polynomial).

The same questions are posed, with the same structure, for the differential-algebraic specialization: the multivariable differential dimension polynomial $\Phi_\eta$ of a finitely generated extension of a differential field of characteristic zero with a partitioned basic set of derivations, where the invariants are required to be birational invariants of the extension.

References

1. Alexander B. Levin, Gröbner bases with respect to several orderings and multivariable dimension polynomials, Journal of Symbolic Computation 42 (2007), no. 5, 561-578. DOI: 10.1016/j.jsc.2006.05.006, https://doi.org/10.1016/j.jsc.2006.05.006
2. E. R. Kolchin, Differential Algebra and Algebraic Groups, Pure and Applied Mathematics, vol. 54, Academic Press, New York, 1973. Zbl 0264.12102, https://zbmath.org/?q=an:0264.12102
3. E. R. Kolchin, The notion of dimension in the theory of algebraic differential equations, Bulletin of the American Mathematical Society 70 (1964), 570-573. DOI: 10.1090/s0002-9904-1964-11203-9, https://doi.org/10.1090/s0002-9904-1964-11203-9
4. M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, E. V. Pankratiev, Differential and Difference Dimension Polynomials, Kluwer Academic Publishers, Dordrecht, 1999. DOI: 10.1007/978-94-017-1257-6, https://doi.org/10.1007/978-94-017-1257-6
5. I. N. Bernshtein, Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients, Functional Analysis and Its Applications 5 (1971), no. 2, 89-101. DOI: 10.1007/BF01076413, https://doi.org/10.1007/BF01076413
6. Meng Zhou, Franz Winkler, Computing difference-differential dimension polynomials by relative Gröbner bases in difference-differential modules, Journal of Symbolic Computation 43 (2008), no. 10, 726-745. DOI: 10.1016/j.jsc.2008.02.001, https://doi.org/10.1016/j.jsc.2008.02.001
7. Meng Zhou, Guanli Huang, Gröbner bases in difference-differential modules with coefficients in a commutative ring, Science China Mathematics 55 (2012), no. 9, 1961-1970. DOI: 10.1007/s11425-012-4471-2, https://doi.org/10.1007/s11425-012-4471-2
8. Christian Dönch, Alexander Levin, Computation of the Strength of PDEs of Mathematical Physics and their Difference Approximations, preprint (2012). arXiv:1205.6762, https://arxiv.org/abs/1205.6762
9. Alexander Levin, Multivariate Dimension Polynomials of Inversive Difference Field Extensions, in: Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS 2012), Lecture Notes in Computer Science 8372, Springer, 2014, pp. 146-163. DOI: 10.1007/978-3-642-54479-8_7, https://doi.org/10.1007/978-3-642-54479-8_7
10. Alexander Levin, Multivariate difference-differential dimension polynomials and new invariants of difference-differential field extensions, Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation (ISSAC 2013), 2013, pp. 267-274. DOI: 10.1145/2465506.2465521, https://doi.org/10.1145/2465506.2465521
11. Christian Dönch, Characterization of relative Gröbner bases, Journal of Symbolic Computation 55 (2013), 19-29. DOI: 10.1016/j.jsc.2013.03.003, https://doi.org/10.1016/j.jsc.2013.03.003
12. Christian Dönch, Alexander Levin, Bivariate dimension polynomials and new invariants of finitely generated D-modules, International Journal of Algebra and Computation 23 (2013), no. 7, 1625-1651. DOI: 10.1142/S0218196713500409, https://doi.org/10.1142/S0218196713500409
13. Guanli Huang, Meng Zhou, Termination of algorithm for computing relative Gröbner bases and difference differential dimension polynomials, Frontiers of Mathematics in China 10 (2015), no. 3, 635-648. DOI: 10.1007/S11464-015-0439-1, https://doi.org/10.1007/S11464-015-0439-1
14. Christoph Fürst, Günter Landsmann, Computation of Dimension in Filtered Free Modules by Gröbner Reduction, Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation (ISSAC 2015), 2015, pp. 181-188. DOI: 10.1145/2755996.2756680, https://doi.org/10.1145/2755996.2756680
15. Christoph Fürst, Alexander Levin, Relative Reduction and Buchberger's Algorithm in Filtered Free Modules, Mathematics in Computer Science 11 (2017), no. 3-4, 329-339. DOI: 10.1007/s11786-017-0317-1, https://doi.org/10.1007/s11786-017-0317-1
16. Alexander Levin, Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals. The Case of One Translation, Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation (ISSAC 2018), 2018, pp. 255-262. DOI: 10.1145/3208976.3209008, https://doi.org/10.1145/3208976.3209008
17. Alexander Levin, Bivariate Dimension Quasi-polynomials of Difference-Differential Field Extensions with Weighted Basic Operators, Mathematics in Computer Science 13 (2019), no. 1-2, 157-168. DOI: 10.1007/s11786-018-0361-5, https://doi.org/10.1007/s11786-018-0361-5
18. Omar León Sánchez, Estimates for the coefficients of differential dimension polynomials, Mathematics of Computation 88 (2019), no. 320, 2959-2985. DOI: 10.1090/mcom/3429, https://doi.org/10.1090/mcom/3429
19. Alexander Levin, Multivariate Difference-Differential Dimension Polynomials, Mathematics in Computer Science 14 (2020), no. 2, 361-374. DOI: 10.1007/s11786-019-00436-1, https://doi.org/10.1007/s11786-019-00436-1; preprint arXiv:1207.4757, https://arxiv.org/abs/1207.4757
20. Alexander Levin, Some properties of multivariate differential dimension polynomials and their invariants, Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation (ISSAC 2020), 2020, pp. 320-327. DOI: 10.1145/3373207.3404013, https://doi.org/10.1145/3373207.3404013
21. Günter Landsmann, Christoph Fürst, An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles, American Journal of Applied Mathematics 9 (2021), no. 4. DOI: 10.11648/J.AJAM.20210904.14, https://doi.org/10.11648/J.AJAM.20210904.14
22. Alexander Levin, Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals. The case of one translation, Journal of Symbolic Computation 102 (2021), 173-188. DOI: 10.1016/j.jsc.2019.10.014, https://doi.org/10.1016/j.jsc.2019.10.014
23. Hamzeh Harfsheno, Abdolali Basiri, Sajjad Rahmany, Generalization of Buchberger's Algorithm with Respect to Several Orderings on Difference Modules, Mathematical Researches 7 (2021), no. 3, 513-526. DOI: 10.52547/mmr.7.3.513, https://doi.org/10.52547/mmr.7.3.513
24. Alexander Levin, Generalized Gröbner Bases and New Properties of Multivariate Difference Dimension Polynomials, Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation (ISSAC 2021), 2021, pp. 273-280. DOI: 10.1145/3452143.3465544, https://doi.org/10.1145/3452143.3465544
25. Alexander Levin, Reduction with Respect to the Effective Order and a New Type of Dimension Polynomials of Difference Modules, Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation (ISSAC 2022), 2022, pp. 55-62. DOI: 10.1145/3476446.3535497, https://doi.org/10.1145/3476446.3535497
26. M. V. Kondratieva, Some Properties of Coefficients Kolchin Dimension Polynomial, preprint (2022). arXiv:2202.02542, https://arxiv.org/abs/2202.02542
27. Alexander Levin, A New Type of Difference Dimension Polynomials, Mathematics in Computer Science 16 (2022), no. 4. DOI: 10.1007/s11786-022-00540-9, https://doi.org/10.1007/s11786-022-00540-9
28. Alexander Levin, Generalized characteristic sets and new multivariate difference dimension polynomials, Applicable Algebra in Engineering, Communication and Computing 35 (2024), no. 1, 31-53. DOI: 10.1007/s00200-023-00628-0, https://doi.org/10.1007/s00200-023-00628-0
29. Alexander Levin, New Difference Gröbner Bases and Bivariate Difference Dimension Polynomials, Mathematics in Computer Science 18 (2024), no. 3. DOI: 10.1007/s11786-024-00589-8, https://doi.org/10.1007/s11786-024-00589-8
30. Alexander Levin, New Multivariate Dimension Polynomials of Inversive Difference Field Extensions, Journal of Mathematical Sciences 284 (2024), no. 4, 508-526. DOI: 10.1007/s10958-024-07367-8, https://doi.org/10.1007/s10958-024-07367-8
31. Alexander Levin, Multivariate Bernstein-type Polynomials of Finitely Generated D-Modules, preprint (2024). arXiv:2404.02069, https://arxiv.org/abs/2404.02069
32. Harry Prieto, Computing the Bernstein polynomial and the Krull-type dimension of finitely generated D-modules, preprint (2024). arXiv:2411.09537, https://arxiv.org/abs/2411.09537
33. Alexander Levin, New dimension polynomials of inversive difference field extensions and inversive difference modules, Journal of Symbolic Computation 134 (2026), article 102521. DOI: 10.1016/j.jsc.2025.102521, https://doi.org/10.1016/j.jsc.2025.102521
34. Fritz Grimpen, Matthias Orth, Anastasios Stefanou, Relative Gröbner bases of modules and applications in persistence theory, preprint (2026). arXiv:2603.27149, https://arxiv.org/abs/2603.27149