arXiv subjects

Timur R. Seifullin

Publications and source records attributed to Timur R. Seifullin.

6 recordsLinked to original sources

arXiv2020

Symmetric polynomials and exterior power of a polynomial ring in one variable

In this article we consider the exterior power and the symmetric tensors of the polynomial ring in one variable. The structure of an associative semigraded algebra of this polynomial ring induces on the symmetric tensors the structure of an associative semigraded algebra, and on the exterior power induces structure of a semigraded module over semigraded algebra of symmetric tensors. The algebra of symmetric polynomials is isomorphic to the algebra of the symmetric tensors of polynomial ring in one variables. We obtained the explicit expression for symmetric polynomials via elementary symmetric polynomials and the explicit expression for elements of the exterior power via elementary symmetric polynomials and elements of the exterior power of the lower polynomial degree.

math.AC↗

arXiv2012

Koszul complexes of embedded systems of polynomials and duality

The object of the paper is the dependence of Koszul complexes and dependence of dual Koszul complexes of two systems of non-homogeneous polynomials, when one system is a part of other system, in connection with the duality in a Koszul complex established by author earlier. Whence, the dependence of Koszul complexes and dependence of dual Koszul complexes follow when one system is linearly expressed through other system. Obtaned results are used in the proof of homotopic equivalence, formulated earlier by the author, of the Koszul complex and dual Koszul complex of a system of non-homogeneous polynomials, what happens when the ideal of these polynomials is 0-dimensional.

math.AC↗

arXiv2012

Homology of the Koszul complex of a system of polynomial equations

For a system of non-homogeneous polynomials it was constructed explicit complex morphism of a dual complex to the Koszul complex into the Koszul complex. If the ideal of these polynomials is 0-dimensional, then this mapping is a homotopic equivalence, thus in this case it was obtained explicit duality of the Koszul complex.

math.AC↗

arXiv2009

Extension of bounded root functionals of a system of polynomial equations

The notion of a root functional of a system of polynomials or ideal of polynomials is a generalization of the notion of a root, in particular, for a multiple root. A root functional is a linear functional that is defined on a polynomial ring and annuls the ideal of polynomials. A bounded root functional is a functional that annuls d-th component of the ideal in some filtration in this ideal. The paper consider bounded root functionals and their extension operation for a system of polynomial equation at which the number of equations is equal to the number of unknows. The extension operation has connection with the multivariate Bezoutian construction.

math.AC↗

arXiv2008

Determination of the basis of the space of all root functionals of a system of polynomial equations and of the basis of its ideal by the operation of the extension of bounded root functionals

It is proposed the algorithm that find a basis of the ideal and a basis of the space of all root functionals by using the extension operation for bounded root functionals, when the number of polynomials is equal to the number of variables, if it is known that the ideal of polynomials is 0-dimensional. The asyptotic complexity of this algorithm is d^{O(n)} operations, where n is the number of polynomials and the number of variables, d is the maximal degree of polynomials. The extension operation has connection with the multivariate Bezoutian construction.

math.AG↗

arXiv2008

Continuation of root functionals of a system of polynomial equations and the reduction of polynomials modulo its ideal

The notion of a root functional of polynomials is a generalization of the notion of a root for a multiple root. A root functional is a linear functional that is defined on a polynomial ring and annuls the ideal of a system of polynomials. A bounded root functional is a functional that annuls d-th component of the ideal in some filtration in this ideal. It was constructed the operation of continuation of root functionals and the operation of reduction of polynomials modulo the ideal on the basis of the extension operation for bounded root functionals when the number of polynomials is equal to the number of variables and the ideal of polynomials is 0-dimensional. The extension operation has connection with the multivariate Bezoutian construction.

math.AC↗