Related papers: Algorithms for computing mixed multiplicities, mix…
We give an explicit geometric way to build mixed-integer programming (MIP) formulations for unions of polyhedra. The construction is simply described in terms of spanning hyperplanes in an r-dimensional linear space. The resulting MIP…
A representation of finite fields that has proved useful when implementing finite field arithmetic in hardware is based on an isomorphism between subrings and fields. In this paper, we present an unified formulation for multiplication in…
In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of…
Using a certain Pick-type formula for the mixed volume of Cayley polyhedra, we compute the multiplicity of the isolated common zero of the maximal minors for a matrix of generic homogeneous polynomials of given degrees.
The Minkowski mixed volume of $n$ subpolytopes $D_1, \dots, D_n$ of a polytope $P \subset {\mathbb R}^n$ clearly does not exceed the normalized volume $n! \text{Vol}(P)$. Equality holds if and only if the subpolytopes are interlaced, i.e.,…
We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation…
In this paper we describe the equations defining the multi-Rees algebra $R[I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where $R$ is a Noetherian ring and the ideals are generated by subsets of a fixed weak regular sequence.
Let $S$ be a finitely generated standard multigraded algebra over an Artinian local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper answers to the question when mixed multiplicities of $M$ are positive and…
Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we…
We prove a theorem on the intersection theory over a Noetherian local ring $R$, which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and…
We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the…
We introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some…
The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, has been developed by Cutkosky, Sarkar and Srinivasan. The objective of this article is to generalise a…
One of the biggest open problems in computational algebra is the design of efficient algorithms for Gr{\"o}bner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of…
This paper investigates the relationship between multiplicities and the degree sequence of ideals in graded algebras, gives multiplicity equations of graded rings via the degree sequence of ideals, and characterizes mixed multiplicities and…
This article discusses the geometric application of the method of multiplier ideal sheaves. It first briefly describes its application to effective problems in algebraic geometry and then presents and explains its application to the…
Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer…
The motivation for this paper are computer calculations of complete lists of weight systems of quasihomogeneous polynomials with isolated singularity at 0 up to rather large Milnor numbers. We review combinatorial characterizations of such…
We give an effective method to determine the multiplier ideals and jumping numbers associated with a curve singularity $C$ in a smooth surface. We characterize the multiplier ideals in terms of certain Newton polygons, generalizing a…
In this paper, we describe a new hybrid algorithm for computing all singular triplets above a given threshold and provide its implementation in MATLAB/Octave and R. The high performance of our codes and ease at which they can be used,…