Toroidalization of generating sequences in dimension two function fields of positive characteristic

Toroidalization of generating sequences in dimension two function fields

Let k be an algebraically closed field of characteristic 0 and let K*/K be a finite extension of algebraic function fields of transcendence degree 2 over k. Let v* be a k-valuation of K* with valuation ring V* and let v be the restriction of v* to K. Suppose R --> S is an extension of algebraic regular local rings with quotient fields K and K*, respectively, such that V* dominates S and S dominates R. We prove that there exist sequences of quadratic transforms R --> R' and S --> S' along v* such that S' dominates R' and the map between generating sequences of v and v* in R' and S', respectivel…

A characterization of finite factorization positive monoids

We provide a characterization of the positive monoids (i.e., additive submonoids of the nonnegative real numbers) that satisfy the finite factorization property. As a result, we establish that positive monoids with well-ordered generating sets satisfy the finite factorization property, while positive monoids with co-well-ordered generating sets satisfy this property if and only if they satisfy the bounded factorization property.

On the construction of valuations and generating sequences on hypersurface singularities

Suppose that (K, $ν$) is a valued field, f (z) $\in$ K[z] is a unitary and irreducible polynomial and (L, $ω$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $ν$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $ω$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $ω$ as an extension of the associated graded ring of A for the filtration defined by $ν$. In parti…

Curves generating extremal rays in blowups of weighted projective planes

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is algebraically related to other curves in both classes; this allows us to find their defining equations inductively. For each curve in our classification, we consider a family of blowups in which th…

阿基米德二次模中严格正多项式的证书计算

Computing Certificates of Strictly Positive Polynomials in Archimedean Quadratic Modules

本研究结合推广的Averkov构造与Lasserre方法，提出阿基米德二次模中严格正多项式证书的构造性计算方法，可适用于全局情形，在RealCertify难以处理的示例中成功生成证书，且更易理解实现。

Short rational generating functions for lattice point problems

We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative…