Related papers: Solving sums of squares in global fields
To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add…
We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and…
For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix…
Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper…
A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and…
We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative…
Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptography.…
The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally…
In this paper, we derive an explicit combinatorial formula for the number of $k$-subset sums of quadratic residues over finite fields.
We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a…
We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…
In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using…
A new general decomposition theory inspired from modular graph decomposition is presented. This helps unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs,…
This paper introduces a notion of decomposition and completion of sum-of-squares (SOS) matrices. We show that a subset of sparse SOS matrices with chordal sparsity patterns can be equivalently decomposed into a sum of multiple SOS matrices…
A \emph{square} is a finite non-empty word consisting of two identical adjacent blocks. A word is \emph{square-free} if it does not contain a square as a factor. In any finite word one may delete the repeated block of a square, obtaining…
Geometric modeling by constraints leads to large systems of algebraic equations. This paper studies bipartite graphs underlaid by systems of equations. It shows how these graphs make possible to polynomially decompose these systems into…
The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem.…
Building an infinite square-free word by appending one letter at a time while simultaneously avoiding the creation of squares is most likely to fail. When the alphabet has two letters this approach is impossible. When the alphabet has three…
For each odd prime power q, and each integer k, we determine the sum of the k-th powers of all elements x in F_q for which both x and x+1 are squares in F_q^*. We also solve the analogous problem when one or both of x and x+1 is a…
General methods for the construction of magic squares of any order have been searched for centuries. There have been several standard strategies for this purpose, such as the knight movement, or the construction of bordered magic squares,…