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In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear…
We study ideals which are generated by monomials of degree $d$ in the polynomial ring in $n$ variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese…
In nice cases, a zero-dimensional complete intersection ideal over a field of characteristic zero has a Shape Lemma. There are also cases where the ideal is generated by the resultant and first subresultant polynomials of the generators.…
We study a class of combinatorially-defined polynomial ideals which are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the…
Polyomino ideals, defined as the ideals generated by the inner $2$-minors of a polyomino, are a class of binomial ideals whose algebraic properties are closely related to the combinatorial structure of the underlying polyomino. We provide a…
A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix. whose elements are the coefficients of the input polynomials…
Let $R$ be a commutative Noetherian ring, $I$ an ideal, $M$ and $N$ finitely generated $R$-modules. Assume $V(I)\cap Supp(M)\cap Supp(N)$ consists of finitely many maximal ideals and let ${\l}(\e^i(N/I^nN,M))$ denote the length of…
We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.
Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphisms between the special fiber…
We characterize the class of ideals of a polynomial ring such that the hilbert series of their graded local cohomology modules is maximal.
We specify a small set, consisting of $O(d(\log\log d)^2)$ points, that intersects the basins under Newton's method of \emph{all} roots of \emph{all} (suitably normalized) complex polynomials of fixed degrees $d$, with arbitrarily high…
A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order…
A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyze criteria for a homogeneous polynomial to be decomposable as a direct…
A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…
We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume…
By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper…
Consider a finite system of non-strict polynomial inequalities with solution set $S\subseteq\mathbb R^n$. Its Lasserre relaxation of degree $d$ is a certain natural linear matrix inequality in the original variables and one additional…
This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main result in…
For a positive integer $d$, a non-negative integer $n$ and a non-negative integer $h\leq n$, we study the number $C_{n}^{(d)}$ of principal ideals; and the number $C_{n,h}^{(d)}$ of principal ideals generated by an element of rank $h$, in…