Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2404.10377

Separating Variables in Bivariate Polynomial Ideals: the Local Case

Manfred Buchacher
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a semi-algorithm which for any irreducible $p\in\mathbb{K}[x,y]$ finds all elements of $\mathbb{K}(x) + \mathbb{K}(y)$ that are of the form $qp$ for some $q\in\mathbb{K}(x,y)$ whose denominator is not divisible by $p$.

Explore related subjects

math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Manfred Buchacher. 2025-06-02. Separating Variables in Bivariate Polynomial Ideals: the Local Case. https://arxiv.org/abs/2404.10377

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Unified Common-Root and Interpolation Bounds Based on Leading Monomial Data

For general fields the footprint bound from Gröbner basis theory estimates the number of common affine roots of any set of multivariate polynomials using information on their leading monomials. In this paper we develop an interpolation bound with a similar flavor extending a previously known result for only a single polynomial to any prescribed number of polynomials. Surprisingly, our interpolation theorem and the footprint bound can be shown to be two sides of the same coin, solving similar problems, but for dual spaces. As discussed the footprint bound compares well with the improved Alon-Füredi bound and for finite fields the presented interpolation theorem is sharp. Our work can be viewed as a comment to a question raised by Tao in [Tao, 2014]

math.AC↗
arXiv2026

Remarks on some Homological Problems regarding Infinite Integral Extensions

Let $R$ be an excellent local domain. $R$ is said to be $NBIM$ if $Tor_{i}^{R}(R^{+}, k) = 0$ for some $i\geq d:=\dim(R)$. Bhatt, Iyengar, and Ma ask if equi-characteristic zero $NBIM$ rings are regular. If $R$ is of positive characteristic, Asgharzadeh and Mahdavi conjecture that $Ext^{i}_{R}(k,R^{\infty}) = 0$ for some $i>d$ implies that $R$ is regular. It is an open question whether $R^{+}$ and $R^{\infty}$ are $\mathfrak{m}$-adically idealwise separated in positive characteristic, a condition from the `local criterion of flatness'. These are analogues of Kunz's theorem and intimately related to the homological conjectures and singularities in algebraic geometry. We apply a result of Avramov, Hochster, Iyengar, and Yao on contracting endomorphisms to make progress on the first two. We observe that it implies toric $NBIM$ rings are regular and solves the conjecture for $F$-pure rings. These improvements are inaccessible by previous techniques and give new and simple proofs of earlier results. In mixed characteristic, we show several linked results for perfectoid-pure rings. We show the third statement when there is $R\rightarrow S$ finite and flat on the punctured spectrum and $S$ is regular, this uses Cohen-Macaulayness of $S^{+}$.

math.AC↗
arXiv2026

Descent along flat composed with radicalization

We study the following general situation. Let $R\to S$ be a finite flat morphism of regular rings of zero (resp. prime) characteristic. For $P\in Spec R$, put $A=R/P, C=S/PS,$ and $B=S/\sqrt{PS}=C_{\mathrm{red}}.$ The basic question is whether a property of $B$ forces the same property of $A$. The main point is that ordinary finite-flat descent applies naturally to $A\to C$, whereas the passage $C\to C_{\mathrm{red}}$ may destroy nilpotent information.

math.AC↗