Definability of primality for radical differential ideals and the generalized Ritt problem (Harrison-Trainor–Klys–Moosa Conjecture 1.1)

For systems of polynomial differential equations over a characteristic-zero differential field, is there a uniform bound, depending only on the number of derivations, the number of indeterminates, and the degrees and orders of the input, such that primality of the radical differential ideal generated by the system can be certified by checking the multiplicative primality condition only on differential polynomials of bounded degree and order? This is the definability-of-primality conjecture posed by Harrison-Trainor, Klys, and Moosa (J. Algebra, 2012), who proved it equivalent to four other natural existence-of-bounds problems in differential algebra, to a nonstandard statement about internal differential polynomial rings, and — via the equivalence web of Golubitsky–Kondratieva–Ovchinnikov — to the generalized Ritt problem of deciding inclusion of prime differential ideals presented by characteristic sets. Later work (proof mining, definability of Kolchin polynomials, algorithm-to-bound transfer) has made the surrounding partial results effective but left this core open in both directions.

Background

Differential algebra treats systems of polynomial differential equations algebraically. A differential field (or Δ-field) is a field $k$ of characteristic zero equipped with finitely many commuting derivations $\Delta=\{\delta_1,\dots,\delta_m\}$ ; the ring of differential polynomials $k\{X\}$ in indeterminates $X=(X_1,\dots,X_n)$ is the commutative polynomial ring in the infinitely many algebraic indeterminates $\theta X_i$ , where $\theta$ ranges over all derivative operators $\delta_1^{e_1}\cdots\delta_m^{e_m}$ , equipped with the induced derivations. A differential ideal is an ideal closed under the derivations; for a finite set $S\subset k\{X\}$ , $\{S\}$ denotes the radical differential ideal generated by $S$ , and it is prime when the quotient $k\{X\}/\{S\}$ is an integral domain. The degree of a differential polynomial is its total degree as a commutative polynomial in the variables $\theta X_i$ , and its order is the largest $e_1+\cdots+e_m$ among the derivative operators occurring in it. Geometrically, over a differentially closed field (the model companion of the theory of differential fields, which admits quantifier elimination), the zero set of $S$ is an irreducible Kolchin-closed set — a closed set for the differential-algebraic (Kolchin) topology — exactly when $\{S\}$ is prime; thus primality of $\{S\}$ is the algebraic counterpart of irreducibility of a differential-algebraic variety, and asking whether the set of irreducible members of a differential-algebraic family is itself definable amounts to asking whether primality can be checked using only bounded objects.

For ordinary polynomial rings without derivations the analogous question has a positive answer from constructive commutative algebra, with a clean nonstandard proof by van den Dries and Schmidt (Inventiones Mathematicae, 1984): there is a bound, depending only on the degrees of the generators, such that primality of a radical ideal is certified by checking the multiplicative condition on polynomial pairs of bounded degree — which is why irreducibility is definable in algebraic families of varieties. Harrison-Trainor, Klys, and Moosa extended this ultraproduct method to differential polynomial rings. In their nonstandard setting one passes to an ultraproduct $\prod_\mathcal{U} k_r\{X\}$ along a nonprincipal ultrafilter $\mathcal{U}$ ; its elements, the internal differential polynomials $K\{X\}_{\mathrm{int}}$ , are sequences of differential polynomials identified modulo $\mathcal{U}$ and can involve derivatives of infinite order (such as $\delta^\omega X$ ), which is the source of both the power and the difficulties of the method. A characteristic set, the other key notion, is a finite autoreduced subset of a prime differential ideal $P$ of least possible rank (autoreduced means no element's leading derivative variable is a derivative of another's); it is the standard finite presentation of $P$ in algorithms of the Ritt–Kolchin tradition.

Harrison-Trainor, Klys, and Moosa prove a one-sided bound (their Theorem 5.4: for each $d$ there is $r$ such that primality is equivalent to the bounded condition with only one of the two factors constrained to degree and order $\le r$ ), bounds for characteristic sets of minimal prime differential ideals (Theorem 6.1) and for the differential Nullstellensatz (Theorem 6.3), and — the subject of this problem — that the full two-sided bound conjecture is equivalent to a family of neighboring open problems: statement (A) about internal differential polynomial rings; the existence of bounds for generators of a prime differential ideal presented by a characteristic set; bounds for checking inclusion of prime differential ideals from characteristic sets; bounds for distinguishing minimal prime differential ideals; and definability of zero-divisibility. The algorithmic analogue of the inclusion problem is the generalized Ritt problem of Golubitsky–Kondratieva–Ovchinnikov: is there an algorithm which, given characteristic sets of two prime differential ideals, decides whether one is contained in the other? Later work in proof mining and model theory has made several of the surrounding results explicit and effective, but the central conjecture remains unresolved.

Problem Statement

Does there exist, for every $d$ , $n$ , and $m$ , a bound $r=r(d,n,m)$ such that for every field $k$ of characteristic zero equipped with $m$ commuting derivations $\Delta$ and every finite set $S$ of $\Delta$ -polynomials in $n$ indeterminates over $k$ , all of degree and order $\le d$ , the following are equivalent? (i) $\{S\}$ , the radical $\Delta$ -ideal generated by $S$ , is prime; (ii) $\{S\}$ is proper, and for all $\Delta$ -polynomials $f$ and $g$ over $k$ of degree and order $\le r$ , if $fg\in\{S\}$ then $f\in\{S\}$ or $g\in\{S\}$ .

This is Conjecture 1.1 (Definability of primality) of Harrison-Trainor–Klys–Moosa. By the equivalences proved in that paper and in the surrounding literature, an answer in either direction — positive or negative — is simultaneously an answer to each of the following: statement (A), that for every internal $\Delta$ -field $K$ and internal $S\subset K\{X\}$ , primeness of $\{S\}_{\mathrm{int}}\cap K\{X\}$ in $K\{X\}$ implies primeness of $\{S\}_{\mathrm{int}}$ in $K\{X\}_{\mathrm{int}}$ ; the existence of bounds for generators of a prime $\Delta$ -ideal presented by a characteristic set; the existence of bounds for checking inclusion of prime $\Delta$ -ideals from characteristic sets; the existence of bounds for distinguishing minimal prime $\Delta$ -ideals; definability of zero-divisibility modulo a radical $\Delta$ -ideal; and decidability of the generalized Ritt problem (given characteristic sets of two prime differential ideals, decide whether one contains the other) over computable base fields. Resolving any one of these equivalent forms, positively or negatively, resolves the conjecture.

References

Matthew Harrison-Trainor, Jack Klys, Rahim Moosa, 'Nonstandard methods for bounds in differential polynomial rings', Journal of Algebra 360 (2012) 71–86. DOI: 10.1016/j.jalgebra.2012.03.013; arXiv:1105.0600. https://doi.org/10.1016/j.jalgebra.2012.03.013
Oleg Golubitsky, Marina Kondratieva, Alexey Ovchinnikov, 'On the generalized Ritt problem as a computational problem' (the arXiv preprint version is titled with the British spelling 'generalised'), Journal of Mathematical Sciences 163 (2009) 515–522. DOI: 10.1007/s10958-009-9689-3; arXiv:0809.1128. https://doi.org/10.1007/s10958-009-9689-3
James Freitag, Omar León Sánchez, Wei Li, 'Effective definability of Kolchin polynomials', Proceedings of the American Mathematical Society 148 (2020) 1455–1466. DOI: 10.1090/proc/14869; arXiv:1806.02060. https://doi.org/10.1090/proc/14869
William Simmons, Henry Towsner, 'Proof mining and effective bounds in differential polynomial rings', Advances in Mathematics 343 (2019) 567–623. DOI: 10.1016/j.aim.2018.11.026; arXiv:1609.07509. https://doi.org/10.1016/j.aim.2018.11.026
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L. van den Dries, K. Schmidt, 'Bounds in the theory of polynomial rings over fields. A nonstandard approach', Inventiones mathematicae 76 (1984) 77–91. DOI: 10.1007/BF01388493. https://doi.org/10.1007/BF01388493
Gleb Pogudin, 'Prime differential nilalgebra exists', Moscow University Mathematics Bulletin 69 (2014) 33–36. DOI: 10.3103/S0027132214010069; arXiv:1304.5765 (2013). https://doi.org/10.3103/S0027132214010069