Atsushi Ito, Makoto Miura
In this paper, we study a relation between Seshadri constants and degrees of defining polynomials. In particular, we compute the Seshadri constants on Fano varieties obtained as complete intersections in rational homogeneous spaces of Picard number one.
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
We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.
Seshadri constants and Gromov widths of toric varieties associated with generalized permutohedra
本文研究极化的环面簇的Seshadri常数和Gromov宽度，其矩多面体为广义置换体，证明两个不变量等于矩多面体的格宽度，并给出基于定义子模函数的显式公式。
Jets-separation thresholds, Seshadri constants and higher Gauss-Wahl maps on abelian varieties
本文在极化阿贝尔簇上定义闭子概形的消失阈值与射流分离阈值，并利用Fourier-Mukai方法建立其与高阶Gauss-Wahl映射满射性及Seshadri常数之间的联系，给出满射性判别准则。
Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces
本文研究Hirzebruch曲面上由无穷远处非正除子赋值定义的旗，计算相应Seshadri型常数并显式确定Newton-Okounkov体的顶点，证明其为四边形或三角形。
Let $f_{1}, \ldots, f_{k}$ be polynomials defining an algebraic set in affine $n$-space over a finite field. Suppose $k>n$. We prove that there exists a system of polynomials $g_{1}, \ldots, g_{n}$, each being a linear combination with scalar coefficients of $f_{1}, \ldots, f_{k}$, defining the same algebraic set. In particular, one reduces the number of equations without increasing the total degree. We also have the corresponding result for systems of homogeneous polynomials defining algebraic sets in projective spaces.
Let $P$ be a bounded convex subset of $\mathbb R^n$ of positive volume. Denote the smallest degree of a polynomial $p(X_1,\dots,X_n)$ vanishing on $P\cap\mathbb Z^n$ by $r_P$ and denote the smallest number $u\geq0$ such that every function on $P\cap\mathbb Z^n$ can be interpolated by a polynomial of degree at most $u$ by $s_P$. We show that the values $(r_{d\cdot P}-1)/d$ and $s_{d\cdot P}/d$ for dilates $d\cdot P$ converge from below to some numbers $v_P,w_P>0$ as $d$ goes to infinity. The limits satisfy $v_P^{n-1}w_P \leq n!\cdot\operatorname{vol}(P)$. When $P$ is a triangle in the plane, we…
Strong monodromy conjecture for defining polynomials of projective hypersurfaces having only weighted homogeneous isolated singularities
研究显示，在特定条件下，定义多项式的强monodromy猜想可通过已知结果和Denef-Loeser公式推导，且在某些情况下可能不存在反例。
We describe a set of defining relations of the tame automorphism group TA_3(F) of the polynomial algebra F[x_1,x_2,x_3] in variables x_1,x_2,x_3 over an arbitrary field F of characteristic 0.
In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$ represents the support function of this set. Our aim is to examine properties of the degree of the defining polynomial of this family of curves in terms of the degree of $p$. Thanks to the theory of elimination, we compute the total degree and the partial degrees of this polynomial, and we solve in addition a question raised by Rabinowitz in \cite{Rabi} on th…
This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund-Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold.
This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero "obstruction" polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define…
Polynomials of small slice rank and strength
本文研究可约多项式簇割簇的定义方程，编码多项式强度与切片秩概念，提出三项贡献：推广 Ruppert 方程至任意变量、基于偏导数合冲构造小强度行列式方程，并证明三次型切片秩二由14变量泛线性截面决定。
Let $A$ be a commutative Noetherian ring of dimension $n$ ($n \ge 3$). Let $I$ be a local complete intersection ideal in $A[T]$ of height $n$. Suppose $I/{I^2}$ is free ${A[T]}/I$-module of rank $n$ and $({A[T]}/I)$ is torsion in $K_0(A[T])$. It is proved in this paper that $I$ is a set theoretic complete intersection ideal in $A[T]$ if one of the following conditions holds: (1) $n$ $\ge 5$, odd; (2) $n$ is even, and $A$ contains the field of rational numbers; (3) $n = 3$, and $A$ contains the field of rational numbers.
Given a complex algebraic hypersurface~$H$, we introduce a polyhedral complex which is a subset of the Newton polytope of the defining polynomial for~$H$ and enjoys the key topological and combinatorial properties of the amoeba of~$H.$ We provide an explicit formula for this polyhedral complex in the case when the spine of the amoeba is dual to a triangulation of the Newton polytope of the defining polynomial. In particular, this yields a description of the polyhedral complex when the hypersurface is optimal.
Given a polynomial $f$ and a semi-algebraic set $S$, we provide a symbolic algorithm to find the equations and inequalities defining a semi-algebraic set $Q$ which is identical to the closure of the image of $S$ under $f$, i.e., \begin{equation} Q=\overline{f(S)}\,. \end{equation} Consequently, every polynomial optimization problem whose optimum value is finite has an equivalent form with attained optimum value, i.e., \begin{equation} \min \limits_{t\in Q} t =\inf\limits_{x\in S} f(x) \end{equation} whenever the right-hand side is finite. Given $d$ as the upper bound on the degrees of $f$ and …
Let $I$ be the ideal of relations between the leading terms of the polynomials defining an automorphism of $K^n$. In this paper, we prove the existence of a locally nilpotent derivation which preserves $I$. Moreover, if $I$ is principal, i.e. $I=(R)$, we compute an upper bound for $°_2(R)$ for some degree function $°_2$ defined by the automorphism. As applications, we determine all the principal ideals of relations for automorphisms of $K^3$ and deduce two elementary proofs of the Jung-van der Kulk Theorem about the tameness of automorphisms of $K^{2}$.
Consider a height two ideal, $I$, which is minimally generated by $m$ homogeneous forms of degree $d$ in the polynomial ring $R=k[x,y]$. Suppose that one column in the homogeneous presenting matrix $\f$ of $I$ has entries of degree $n$ and all of the other entries of $\f$ are linear. We identify an explicit generating set for the ideal $\Cal A$ which defines the Rees algebra $\Cal R=R[It]$; so $\Cal R=S/\Cal A$ for the polynomial ring $S=R[T_1,...,T_m]$. We resolve $\Cal R$ as an $S$-module and $I^s$ as an $R$-module, for all powers $s$. The proof uses the homogeneous coordinate ring, $A=S/H$,…
In this paper we consider the Newton polygons of $L$-functions coming from additive exponential sums associated to a polynomial over a finite field $\F_q$. These polygons define a stratification of the space of polynomials of fixed degree. We determine the open stratum: we give the generic Newton polygon for polynomials of degree $d\geq 2$ when the characteristic $p$ is greater than 3d, and the Hasse polynomial, i.e. the equation defining the hypersurface complementary to the open stratum.
Paving Matroids: Defining Equations and Associated Varieties
本文研究铺垫型矩阵的实现空间及其闭包，引入三种生成关联理想多项式的不同方法，并计算特定铺垫型矩阵的定义方程，分析点线配置的电路变分分解及实例应用。
We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining a nodal hypersurface. The result gives information on the position of the singularities of a nodal hypersurface expressed in terms of defects or superabundances. The case of Chebyshev hypersurfaces is considered as a test for this result and leads to a potentially infinite family of nodal hypersurfaces having nontrivial Alexander polynomials.
We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved in the complexity class PSPACE: (I) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] defining a variety of dimension <=0 in C^n, find all solutions in Z^n of f_1=...=f_m=0. (II) For a given polynomial f in Z[v,x,y] defining an irreducible nonsingular non-ruled surface in C^3, decide the sentence ``\exists v \forall x \exists y such that f(v,x,y)=0?'' quantified over N. Better still, we show that the truth of the Generalized Rie…
We present another example of a 3-variable polynomial defining a K3-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet L-series.
We show that, for a system of univariate polynomials given in sparse encoding, we can compute a single polynomial defining the same zero set, in time quasi-linear in the logarithm of the degree. In particular, it is possible to determine whether such a system of polynomials does have a zero in time quasi-linear in the logarithm of the degree. The underlying algorithm relies on a result of Bombieri and Zannier on multiplicatively dependent points in subvarieties of an algebraic torus. We also present the following conditional partial extension to the higher dimensional setting. Assume that the …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。