Finite decomposition of P[σ]-perfect ideals into P[σ]-prime ideals

In toric P-difference varieties (Sci. China Math. 2020), Jie Wang introduced the P[σ]-polynomial ring, whose monomials carry exponent polynomials with positive leading coefficient — a class intermediate between ordinary difference polynomials and Laurent difference polynomials — and proved that every P[σ]-perfect ideal is an intersection of P[σ]-prime ideals, in bijection with the subvarieties of affine P[σ]-space. Remark 3.6 of that paper leaves open whether finitely many P[σ]-prime ideals always suffice: the P[σ]-analogue of the classical fact that every perfect σ-ideal is a finite intersection of σ-prime ideals, equivalent to the Ritt–Raudenbush basis theorem.

Background

Let $k$ be a difference field ($\sigma$-field), i.e. here a field of characteristic zero equipped with a ring endomorphism $\sigma$, and let $y_1,\dots,y_m$ be difference indeterminates. The ordinary difference polynomial ring $k\{y_1,\dots,y_m\}$ is the polynomial ring in the variables $y_i,\sigma(y_i),\sigma^2(y_i),\dots$. Writing $x$ for a placeholder that records the action of $\sigma$, for $g=\sum_i c_i x^i\in\mathbb{Z}[x]$ and an element $a$ of a $\sigma$-ring set $a^g=\prod_i\sigma^i(a)^{c_i}$; ordinary difference monomials then have exponent vectors in $\mathbb{N}[x]$. A $\sigma$-ideal is an ideal closed under $\sigma$; it is perfect if $a^g\in I$ with $g\in\mathbb{N}[x]\setminus\{0\}$ implies $a\in I$; it is $\sigma$-prime if it is reflexive ($\sigma(a)\in I$ implies $a\in I$) and prime as an ordinary ideal. The classical Ritt–Raudenbush basis theorem states that every perfect $\sigma$-ideal of $k\{y_1,\dots,y_m\}$ is finitely generated as a perfect ideal, and consequently every perfect $\sigma$-ideal is a finite intersection of $\sigma$-prime ideals; geometrically, every affine difference variety has finitely many irreducible components.

To build a divisor theory for toric difference varieties (introduced by Gao–Huang–Wang–Yuan), Wang enlarged the class of admissible difference monomials. Order $\mathbb{Z}[x]$ by comparing coefficients from the top degree downward, so that $f>0$ exactly when the leading coefficient of $f$ is positive, and let $P[x]=\{f\in\mathbb{Z}[x]\mid f\geq 0\}$, which consists of $0$ together with all integer polynomials with positive leading coefficient; thus $\mathbb{N}[x]\subseteq P[x]\subseteq\mathbb{Z}[x]$, closed under addition and multiplication. The $P[\sigma]$-polynomial ring is $k\{y_1,\dots,y_m\}_{P[\sigma]}:=k[y_i^g\mid g\in P[x]]$: its monomials, such as $y^{x-1}=\sigma(y)/y$, allow negative lower-order exponents (Laurent-type difference monomials) while keeping a positivity condition on the leading exponent, so the ring sits between the ordinary difference polynomial ring and the full Laurent difference polynomial ring. For a monomial $Y^u=\prod_i y_i^{u_i}$ with $u\in P[x]^m$, a $\sigma$-ideal $I$ of this ring is a $P[\sigma]$-ideal if $Y^u f\in I$ implies $Y^{gu}f\in I$ for every $g\in P[x]\setminus\{0\}$; it is $P[\sigma]$-perfect if it is simultaneously perfect and a $P[\sigma]$-ideal; and it is $P[\sigma]$-prime if in addition it is prime as an ordinary ideal. Wang proves (Theorem 3.5) that every $P[\sigma]$-perfect ideal is the intersection of all $P[\sigma]$-prime ideals containing it — an a priori infinite intersection — and (Theorem 3.12) that $P[\sigma]$-perfect ideals are in inclusion-reversing bijection with the $P[\sigma]$-subvarieties of affine difference space, so $P[\sigma]$-prime ideals play the role of irreducible components. What is missing is exactly the finiteness of these decompositions: Remark 3.6 states that in the classical case perfect $\sigma$-ideals are finite intersections of $\sigma$-prime ideals (equivalent to their finite generated property), but that in the $P[\sigma]$-case it is still unknown whether a $P[\sigma]$-perfect ideal can be written as an intersection of finitely many $P[\sigma]$-prime ideals. The question is the difference-geometric Noetherianity/finite-component question for this intermediate monomial structure, and it underlies any effective version of the toric $P$-difference geometry developed in the source paper.

Problem Statement

Let $k$ be a $\sigma$-field of characteristic $0$, let $y_1,\dots,y_m$ be difference indeterminates, and let $k\{y_1,\dots,y_m\}_{P[\sigma]}=k[y_i^g\mid g\in P[x]]$ be the $P[\sigma]$-polynomial ring, where $P[x]=\{f\in\mathbb{Z}[x]\mid f\geq 0\}$ for the order on $\mathbb{Z}[x]$ that compares coefficients from the top degree downward, so that $f\geq 0$ exactly when $f=0$ or the leading coefficient of $f$ is positive, and $a^g=\prod_i\sigma^i(a)^{c_i}$ for $g=\sum_i c_i x^i$. Recall that a $\sigma$-ideal $I$ of this ring (an ideal closed under $\sigma$) is a $P[\sigma]$-ideal if $Y^u f\in I$ implies $Y^{gu}f\in I$ for all $g\in P[x]\setminus\{0\}$ and all monomials $Y^u$ with $u\in P[x]^m$; it is $P[\sigma]$-perfect if it is perfect (i.e. $a^g\in I$ with $g\in\mathbb{N}[x]\setminus\{0\}$ implies $a\in I$) and a $P[\sigma]$-ideal; and it is $P[\sigma]$-prime if it is $P[\sigma]$-perfect and prime as an ordinary ideal.

Question (Wang, Remark 3.6): is every $P[\sigma]$-perfect ideal of $k\{y_1,\dots,y_m\}_{P[\sigma]}$ a finite intersection of $P[\sigma]$-prime ideals? Equivalently, in geometric terms via Wang's Theorem 3.12: does every affine $P[\sigma]$-variety decompose into finitely many irreducible $P[\sigma]$-subvarieties?

The claim is that for every $\sigma$-field $k$ of characteristic zero and every finite $m$, every $P[\sigma]$-perfect ideal of $k\{y_1,\dots,y_m\}_{P[\sigma]}$ is a finite intersection of $P[\sigma]$-prime ideals; a counterexample would be a $\sigma$-field $k$ together with a $P[\sigma]$-perfect ideal that is not such a finite intersection. The companion formulation — finite generation of every $P[\sigma]$-perfect ideal as a $P[\sigma]$-perfect ideal, the exact $P[\sigma]$-analogue of the Ritt–Raudenbush basis theorem, to which the classical decomposition statement is equivalent — is the equivalent formulation of this same question in the $P[\sigma]$ setting.

References

1. Jie Wang, Toric P-difference varieties, Science China Mathematics 63(4), 643–670 (2020). DOI: 10.1007/s11425-018-9319-7, https://doi.org/10.1007/s11425-018-9319-7; preprint arXiv:1608.06811, https://arxiv.org/abs/1608.06811 (source paper; see Remark 3.6 and Theorems 3.4, 3.5, 3.12, and Section 8)
2. Xiao-Shan Gao, Zhang Huang, Jie Wang, Chun-Ming Yuan, Toric difference variety, Journal of Systems Science and Complexity 30(1), 173–195 (2017). DOI: 10.1007/s11424-017-6174-9, https://doi.org/10.1007/s11424-017-6174-9; preprint arXiv:1604.01958, https://arxiv.org/abs/1604.01958
3. Jie Wang, Monomial difference ideals, Proceedings of the American Mathematical Society 145(4), 1481–1496 (2017). DOI: 10.1090/proc/13369, https://doi.org/10.1090/proc/13369; preprint arXiv:1601.01148, https://arxiv.org/abs/1601.01148
4. Jie Wang, Finite basis for radical well-mixed difference ideals generated by binomials, Communications in Algebra 46(6), 2589–2599 (2018). DOI: 10.1080/00927872.2017.1392541, https://doi.org/10.1080/00927872.2017.1392541; preprint arXiv:1610.06990, https://arxiv.org/abs/1610.06990
5. Xiao-Shan Gao, Zhang Huang, Chun-Ming Yuan, Binomial difference ideals, Journal of Symbolic Computation 80, 665–706 (2017). DOI: 10.1016/j.jsc.2016.07.029, https://doi.org/10.1016/j.jsc.2016.07.029; preprint arXiv:1603.03987, https://arxiv.org/abs/1603.03987
6. Yu-Ao Chen, Xiao-Shan Gao, Criteria for finite difference Gröbner bases of normal binomial difference ideals, in Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation (ISSAC 2017), 93–100 (2017). DOI: 10.1145/3087604.3087615, https://doi.org/10.1145/3087604.3087615; preprint arXiv:1701.06248, https://arxiv.org/abs/1701.06248 (the arXiv record stores the title with the ASCII spelling "Groebner")
7. Alexey Ovchinnikov, Gleb Pogudin, Thomas Scanlon, Effective difference elimination and nullstellensatz, Journal of the European Mathematical Society 22(8), 2419–2452 (2020). DOI: 10.4171/JEMS/968, https://doi.org/10.4171/JEMS/968; preprint arXiv:1712.01412, https://arxiv.org/abs/1712.01412
8. Michael Wibmer, Finiteness properties of affine difference algebraic groups, International Mathematics Research Notices 2022(1), 506–555 (2022). DOI: 10.1093/imrn/rnaa177, https://doi.org/10.1093/imrn/rnaa177
9. Annette Bachmayr, Michael Wibmer, Trivializing torsors for difference algebraic groups, arXiv:2509.22095 (2025), https://arxiv.org/abs/2509.22095
10. Orla McGrath, Dimension polynomials for affine partial difference algebraic groups, arXiv:2511.20350 (2025), https://arxiv.org/abs/2511.20350