Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of [0, 1]. This implies additive error bounds for rel...

Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is generalized here to arbitrary monomial ideals. It is shown how this duality is naturally expressed by Bass numbers, in their relations to the Betti numbers ...

This paper considers the problem of estimating the parameters of real-valued two-dimensional (2-D) sinusoidal signals observed in colored noise. This problem is a special case of the general problem of estimating the parameters of a real-valued homogeneous random field with mixed spectral distribution from a single observed realization of it. The large sample properties of the least squares (LS...

This report gives an overview of the main ideas in Chapter 9 of the studied book, about the dimension of a variety. After recalling some definitions and basic properties about projective varieties, homogeneous ideals and Gröbner bases, we will begin our study of the dimension of a variety with the observation of a geometric way to define the dimension of a variety defined by a monomial ideal. T...

In recent years, there has been a growing interest in companion matrices. Sparse matrices are well known: every sparse matrix is equivalent to Hessenberg of particular simple type. Recently, Deaett et al. [Electron. J. Linear Algebra, 35:223--247, 2019] started the systematic study nonsparse They proved that nonderogatory, although not necessarily matrix. this paper, which unit described. matri...

The maximal minors of a matrix indeterminates are universal Gröbner basis by theorem Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they not always Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders extensions to bases. These experiments motivated new implementation algorithm which organized in Singular script falls back on ...

Shuffle operads were introduced to forget the symmetric group actions on while preserving all possible operadic compositions. Rewriting methods then applied via shuffle operads: in particular, a notion of Gröbner basis was for with respect total order tree monomials. In this article, we introduce structure polygraphs as categorical model rewriting operads, which generalizes bases approach by re...

Quadratization is a transform of system ODEs with polynomial right-hand side into at most quadratic via the introduction new variables. It has been recently used as pre-processing step for model order reduction methods, so it important to keep number variables small. Several algorithms have designed search quadratization being monomials in original To understand limitations and potential ways i...

let  $r$ be a commutative noetherian ring and $i$ be an ideal of $r$. we say that $i$ satisfies the persistence property if  $mathrm{ass}_r(r/i^k)subseteq mathrm{ass}_r(r/i^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{ass}_r(r/i)$ denotes the set of associated prime ideals of $i$. in this paper, we introduce a class of square-free monomial ideals in the polynomial ring  $r=k[x_1,ld...