arXiv 2103.072822021-03-08math.AGmath.NT

On the last fall degree of Weil descent polynomial systems

Ming-Deh Huang

英文摘要：

Given a polynomial system $\mathcal{F}$ over a finite field $k$ which is not necessarily of dimension zero, we consider the Weil descent $\mathcal{F}'$ of $\mathcal{F}$ over a subfield $k'$. We prove a theorem which relates the last fall degrees of $\mathcal{F}_1$ and $\mathcal{F}'_1$, where the zero set of $\mathcal{F}_1$ corresponds bijectively to the set of $k$-rational points of $\mathcal{F}$, and the zero set of $\mathcal{F}'_1$ is the set of $k'$-rational points of the Weil descent $\mathcal{F}'$. As an application we derive upper bounds on the last fall degree of $\mathcal{F}'_1$ in the case where $\mathcal{F}$ is a set of linearized polynomials.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Descent on elliptic surfaces and arithmetic bounds for the Mordell-Weil rank

We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa's inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.

2021-06-30math.AGmath.NT

Weil's Galois Descent Theorem from a computational point of view

Let ${\mathcal L}/{\mathcal K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over ${\mathcal K}$, that is, for the existence of an algebraic variety $Y$ defined over ${\mathcal K}$ together with a birational isomorphism $R:X \to Y$ defined over ${\mathcal L}$. Weil's proof does not provide a method to construct the birational isomorphism $R.$ The aim of this paper is to give an explicit construction of $R$.

2020-01-02math.AG

带自由算子的代数范畴中的Weil下降函子

The Weil descent functor in the category of algebras with free operators

该研究在对伴随自同态的温和假设下，证明了带自由算子的$\u27e8\b{D}\u27e9$-代数范畴中Weil下降函子的存在性，特别得到了差分代数范畴中Weil下降函子存在这一未见文献报道的新结果。

2023-07-19math.AG

Differential Weil descent

In this short note a differential version of the classical Weil descent is established in all characteristics. This yields a ready-to-deploy tool of differential restriction of scalars for differential varieties over finite differential field extensions.

2020-03-05math.ACmath.AGmath.RA

Differential Weil Descent and Differentially Large Fields

A differential version of the classical Weil descent is established in all characteristics. It yields a theory of differential restriction of scalars for differential varieties over finite differential field extensions. This theory is then used to prove that in characteristic 0, \textit{differential largeness} (a notion introduced here as an analogue to largeness of fields) is preserved under algebraic extensions. This provides many new differential fields with minimal differential closures. A further application is Kolchin-density of rational points in differential algebraic groups defined ov…

2018-07-30math.AGmath.LOmath.RA

代数概形的下降理论

Descent for Algebraic Schemes

本文以初等方式系统阐述了域上代数概形的基本下降定理，涵盖 Grothendieck、Weil 等经典结果。

2024-06-08math.AG

更多相关论文收起更多论文

从算术拓扑角度看降阶理论与等变范畴

An Arithmetic Topology viewpoint on Descent theory and Equivariant Categories

National and Kapodistrian University of Athens（雅典国立卡波蒂斯坦大学）

本文通过算术拓扑视角，统一了降阶理论与等变范畴，建立了算术与拓扑不变量的类比，并展示了其在范畴层面的应用。

2025-12-23math.AGmath.ATmath.CT

On the number of rational points on curves over finite fields with many automorphisms

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the additive group $\Fq$ acts, and Kummer curves of the form $y^{\frac{q-1}{e}}=f(x)$, which have an action of the multiplicative group $\Fq^\star$. In both cases we can remove a $\sqrt{q}$ factor from the Weil bound when $q$ is sufficiently large.

2010-05-27math.AGmath.NT

Explicit two-cover descent for genus 2 curves

Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus $2$ quintic curves over $\mathbb{Q}$ of Mordell-Weil rank $2$ or $3$ whos…

2022-09-02math.AGmath.NT

Mixed Weil cohomologies

We define, for a regular scheme $S$ and a given field of characteristic zero $\KK$, the notion of $\KK$-linear mixed Weil cohomology on smooth $S$-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of $\GG_{m}$ behaves correctly), and Künneth formula. We prove that any mixed Weil cohomology defined on smooth $S$-schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over $S$ to the derived category of the field $\KK$. This implies a finiteness theorem and a Poincaré dualit…

2009-12-10math.AG

Notes on summation polynomials

In these short notes, we will show the following. Let F_q be a finite field and let E/\F_q be an elliptic curve. Let S_r be the rth summation/Semaev polynomial for E. Under an assumption, we show that it is NP-complete to check if S_r evaluates to zero on some input. Unconditionally, we prove a similar result for summation polynomials over singular curves. This suggests limitations in the usage of summation polynomials in for example algorithms to solve the elliptic curve discrete logarithm problem. Assume that q is a power of 2. We show that the Weil descent to F_2 of S_3 for ordinary curves …

2015-06-08math.AGmath.NT

Adelic Descent Theory

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\mathrm{GL}_n(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson's co-simplicial ring of adèles …

2016-02-10math.AGmath.NT

Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in R(x, y)

This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions. As explained by Huisman and Mahe, a given monic squarefree positive polynomial in two variables x and y of degree in y divisible by 4 is a sum of three squares of rational fractions if and only if the jabobian variety of some hyperelliptic curve (associated to P) has an "antineutral" point. Using this criterium, we follow a method developped by Cassels,…

2007-09-13math.AGmath.NT

Bruhat--Tits理论中群概形的美妙嵌入

Wonderful embedding for group schemes in the Bruhat--Tits theory

针对离散赋值亨泽尔域上伴随拟分裂约化群对应的Bruhat--Tits群概形，采用内蕴函子性的Artin--Weil方法变体构造其由大胞腔唯一确定的美妙嵌入，非拟分裂情形通过平展下降实现，搭建了美妙紧化与Bruhat--Tits理论的桥梁。

2025-06-18math.AG