Related papers: A Groebner basis for the secant ideal of the secon…
The Marden theorem of geometry of polynomials and the great Poncelet theorem from projective geometry of conics by their classical beauty occupy very special places. Our main aim is to present a strong and unexpected relationship between…
We call an ideal in a polynomial ring robust if it can be minimally generated by a universal Gr\"obner basis. In this paper we show that robust toric ideals generated by quadrics are essentially determinantal. We then discuss two possible…
In this paper we show that polyomino ideal of a simple polyomino coincides with the toric ideal of a weakly chordal bipartite graph and hence it has a quadratic Gr\"obner basis with respect to a suitable monomial order.
Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial equations whose zeros are uniquely related to Hadamard matrices with one or two circulant cores of a given order. Based on this idea, the cocyclic Hadamard test enable…
Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…
Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of positive integers. Let C be a toric variety in the affine (n+1)-space, defined parametrically by x_0=t^{m_0},...,x_{n}=t^{m_{n}}. In this paper we produce a minimal…
There are several efficient methods to solve linear interval polynomial systems in the context of interval computations, however, the general case of interval polynomial systems is not yet covered as well. In this paper we introduce a new…
Given a parametric polynomial ideal I, the algorithm DISPGB, introduced by the author in 2002, builds up a binary tree describing a dichotomic discussion of the different reduced Groebner bases depending on the values of the parameters,…
In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by…
We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from…
We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs…
In this paper we introduce a working generalization of the theory of Gr\"obner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear…
Generic linkage is used to compute a prime ideal such that the radical of the initial ideal of the prime ideal is equal to the radical of a given codimension two monomial ideal that has a Cohen-Macaulay quotient ring.
We prove the existence of secondary terms of order $X^{5/6}$ in the asymptotic formulas for the average size of the genus number of cubic fields and for the number of cubic fields with a given genus number, establishing improved error…
We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant…
We describe the minimal free resolution of the ideal of $2 \times 2$ subpermanents of a $2 \times n$ generic matrix $M$. In contrast to the case of $2 \times 2$ determinants, the $2 \times 2$ permanents define an ideal which is neither…
The aim of this article is to introduce standard bases of ideals in polynomial rings with respect to a class of orderings which are not necessarily semigroup orderings. Our approach generalises the concept of standard bases with respect to…
We show that the universal Gr\"obner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity…
We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal…
Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $\xi$ to be algebraic in terms…