Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals with several commuting translations

For a non-reflexive prime difference-differential ideal in a ring of difference-differential polynomials with several basic derivations and one translation, Levin (2021) proved the existence of a bivariate Kolchin-type dimension polynomial that exactly expresses the transcendence degrees of all doubly truncated transform/derivative fields, and showed how to compute it by a characteristic-set method with respect to two term orderings. The proof collapses when the single translation is replaced by several commuting translations, because the finite correction set controlling the reduction argument can become infinite; nevertheless the author conjectured that the analogous existence theorem still holds. The problem is to prove or disprove this conjecture: establish (or refute) the existence of a bivariate numerical dimension polynomial for every non-reflexive prime ideal in a difference-differential polynomial ring with two or more commuting translations commuting with the derivations. The polynomial is the exact measure of the degrees of freedom, in the sense of Einstein's strength, of systems of partial algebraic difference-differential equations with several shifts.

Background

A difference-differential field is a field $K$ of characteristic zero equipped with a finite set $\Delta=\{\delta_1,\ldots,\delta_m\}$ of mutually commuting derivations (maps satisfying the Leibniz rule) and a finite set $\Sigma=\{\sigma_1,\ldots,\sigma_h\}$ of pairwise commuting field endomorphisms, called translations or shifts, each commuting with every derivation. The ring of difference-differential ( $\Delta$ - $\Sigma$ -) polynomials $R=K\{y_1,\ldots,y_n\}$ is the polynomial ring over $K$ in the indeterminates $\lambda y_i$ obtained by applying to the $y_i$ all power products $\lambda$ of the derivations and translations with nonnegative integer exponents; it is the natural algebraic home of systems of partial algebraic difference-differential equations. A prime $\Delta$ - $\Sigma$ -ideal $P$ of $R$ is called reflexive if $\gamma(f)\in P$ implies $f\in P$ for every transform $\gamma$ in the semigroup generated by $\Sigma$ ; for a non-reflexive prime ideal the translations induced on the factor ring $R/P$ need not be injective, and this failure invalidates most standard machinery. The reflexive closure $P^*$ of $P$ is the smallest reflexive $\Delta$ - $\Sigma$ -ideal containing $P$ .

Dimension polynomials play for differential and difference algebra the role that Hilbert polynomials play for commutative algebra, going back to Kolchin's differential dimension polynomial for finitely generated differential field extensions. For a prime $\Delta$ - $\Sigma$ -ideal $P$ , one truncates: $R_{rs}=K[\{\tau y_i : \operatorname{ord}_\Delta \tau \le r,\ \operatorname{ord}_\Sigma \tau \le s\}]$ , $P_{rs}=P\cap R_{rs}$ , and $L_{rs}$ is the quotient field of $R_{rs}/P_{rs}$ ; the transcendence degree $\operatorname{tr.deg}_K L_{rs}$ counts the algebraically independent values of the unknown functions and their derivatives of order at most $r$ at all nodes reachable by at most $s$ shifts, so it measures the freedom left by the system. A numerical polynomial is a rational polynomial taking integer values at all sufficiently large integer tuples. The ideal statement of Theorem 4.2 of Levin (2021) is: if $h=1$ (one translation) and $P$ is a (not necessarily reflexive) prime $\Delta$ - $\sigma$ -ideal of $K\{y_1,\ldots,y_n\}$ , then there exists a bivariate numerical polynomial $\psi_P(t_1,t_2)$ , linear in $t_2$ with $\deg_{t_1}\psi_P\le m$ , such that $\psi_P(r,s)=\operatorname{tr.deg}_K L_{rs}$ for all sufficiently large $(r,s)$ . This $\psi_P$ is called the $\Delta$ - $\sigma$ -dimension polynomial of $P$ ; its coefficients carry birational invariants of the associated extension, and $\psi_P$ is the exact value of Einstein's strength of the corresponding system of algebraic difference-differential equations (the number of arbitrary data in its general solution, a concept introduced by Einstein for systems of PDEs in 1953).

The proof uses characteristic sets with respect to two term orderings. A characteristic set, in the sense of the Ritt–Kolchin theory, is an autoreduced set of generators of an ideal in a differential or difference polynomial ring, chosen to have least rank (an autoreduced set is one in which no element is reducible with respect to another; taking one of least rank makes it canonical); it generalizes Gröbner bases to the non-noetherian differential/difference setting and underlies all constructive dimension-polynomial computations. For reflexive prime ideals the corresponding existence theorems with an arbitrary number of translations, and multivariate versions for partitions of the sets of derivations and translations, are classical (Kondratieva–Levin–Mikhalev–Pankratiev 1999; Levin 2008, Theorems 4.2.16–4.2.17; Levin 2020). For non-reflexive prime ideals, existence of the (ordinary difference) dimension polynomial was first obtained non-constructively by model-theoretic arguments of Hrushovski, and Levin's one-translation theorem made it constructive and extended it to the difference-differential setting. In the one-translation proof, the passage from the reflexive closure $P^*$ back to $P$ is controlled by a finite correction set $W$ of terms that are multiples of the leaders of a characteristic set but not of certain entry transforms; with several translations the analogous set $W$ is, generally speaking, infinite, which is exactly where the proof breaks.

Problem Statement

Let $K$ be a difference-differential field of characteristic zero with basic derivations $\Delta=\{\delta_1,\ldots,\delta_m\}$ and a set $\Sigma=\{\sigma_1,\ldots,\sigma_h\}$ , $h\ge 2$ , of pairwise commuting translations (field endomorphisms commuting with one another and with every $\delta_i$ ), let $R=K\{y_1,\ldots,y_n\}$ be the ring of difference-differential ( $\Delta$ - $\Sigma$ -) polynomials over $K$ , and let $P$ be a prime $\Delta$ - $\Sigma$ -ideal of $R$ that is not assumed reflexive (the translations induced on $R/P$ need not be injective). For $r,s\in\mathbb{N}$ let $R_{rs}=K[\{\tau y_i : \operatorname{ord}_\Delta\tau\le r,\ \operatorname{ord}_\Sigma\tau\le s,\ 1\le i\le n\}]$ , let $P_{rs}=P\cap R_{rs}$ , and let $L_{rs}$ be the quotient field of $R_{rs}/P_{rs}$ . Prove or disprove Levin's conjecture, which is the several-translations analog of Theorem 4.2 of Levin (2021): there exists a bivariate numerical polynomial $\psi_P(t_1,t_2)$ such that $\psi_P(r,s)=\operatorname{tr.deg}_K L_{rs}$ for all sufficiently large $(r,s)\in\mathbb{N}^2$ — with, by analogy with the one-translation theorem and the reflexive case, the expected degree bounds $\deg_{t_1}\psi_P\le m$ and $\deg_{t_2}\psi_P\le h$ . A negative answer must exhibit a concrete prime $\Delta$ - $\Sigma$ -ideal $P$ with $h\ge 2$ translations for which $\operatorname{tr.deg}_K L_{rs}$ is not eventually equal to any bivariate numerical polynomial.

References

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 (preprint: arXiv:1909.08957, https://arxiv.org/abs/1909.08957)
Alexander Levin, 'Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals: The Case of One Translation', in: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation (ISSAC 2018), ACM, New York, 2018, 255–262. DOI: 10.1145/3208976.3209008, https://doi.org/10.1145/3208976.3209008
Ehud Hrushovski, 'The Elementary Theory of the Frobenius Automorphisms', preprint, arXiv:math/0406514 (2004; updated version 2012). https://arxiv.org/abs/math/0406514
Alexander Levin, Difference Algebra, Algebra and Applications vol. 8, Springer, New York, 2008. DOI: 10.1007/978-1-4020-6947-5, https://doi.org/10.1007/978-1-4020-6947-5
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
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
Alexander Levin, 'Multivariate Difference–Differential Dimension Polynomials', Mathematics in Computer Science 14 (2020) 361–374. DOI: 10.1007/s11786-019-00436-1, https://doi.org/10.1007/s11786-019-00436-1
Alexander Levin, 'Hilbert-Type Dimension Polynomials of Intermediate Difference-Differential Field Extensions', in: Mathematical Aspects of Computer and Information Sciences (MACIS 2019), Lecture Notes in Computer Science, Springer, Cham, 2020, pp. 64–79. DOI: 10.1007/978-3-030-43120-4_7, https://doi.org/10.1007/978-3-030-43120-4_7 (preprint: arXiv:1911.00875, https://arxiv.org/abs/1911.00875)
Alexander Levin, 'A New Type of Difference Dimension Polynomials', Mathematics in Computer Science 16 (2022), article 20. DOI: 10.1007/s11786-022-00540-9, https://doi.org/10.1007/s11786-022-00540-9
Alexander Levin, 'Generalized characteristic sets and new multivariate difference dimension polynomials', Applicable Algebra in Engineering, Communication and Computing 35 (2024) 31–53. DOI: 10.1007/s00200-023-00628-0, https://doi.org/10.1007/s00200-023-00628-0
Alexander Levin, 'New Difference Gröbner Bases and Bivariate Difference Dimension Polynomials', Mathematics in Computer Science 18 (2024), article 15. DOI: 10.1007/s11786-024-00589-8, https://doi.org/10.1007/s11786-024-00589-8
Alexander Levin, 'New dimension polynomials of inversive difference field extensions and inversive difference modules', Journal of Symbolic Computation 134 (2026) 102521. DOI: 10.1016/j.jsc.2025.102521, https://doi.org/10.1016/j.jsc.2025.102521
Orla McGrath, 'Dimension Polynomials for Affine Partial Difference Algebraic Groups', preprint, arXiv:2511.20350 (2025; v2 2026, to appear in IMRN). https://arxiv.org/abs/2511.20350