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The polarizations of one relation of degree five and two relations of degree six minimally generate the ideal of relations among a minimal generating system of the algebra of multisymmetric polynomials in an arbitrary number of…
In this paper we obtained the formula for the number of irreducible polynomials with degree $n$ over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al.(2003)…
In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
We describe differential invariants of infinite-dimensional algebras being equivalence algebras of some classes of PDE and study structure of these algebras.
We estimate the number of primes represented by a general quadratic polynomial with discriminant $\Delta$, assuming that the corresponding real character is exceptional.
In this paper we give an upper bound, in characteristic 0, for the cohomological dimension of a graded ideal in a polynomial ring such that the quotient has depth at least 3. In positive characteristic the same bound holds true by a…
We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and…
There is a digraph corresponding to every square matrix over $\mathbb{C}$. We generate a recurrence relation using the Laplace expansion to calculate the characteristic, and permanent polynomials of a square matrix. Solving this recurrence…
We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial…
The characteristic polynomial of the effective Hamiltonian for a general model has been discussed. It is found that, compared with the associated energy eigenvalues, this characteristic polynomial generally has better analytical properties…
In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree $2$ polynomials and occur naturally from determinantal conditions. Normality is derived from these results.
We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…
The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various…
We consider finite dimensional representations of the dihedral group $D_{2p}$ over an algebraically closed field of characteristic two where $p$ is an odd integer and study the degrees of generating and separating polynomials in the…
In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when…
We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange \& Hermite…
Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided…
Let f be a real or complex polynomial. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of f.
We develop some basic facts on deformations of exterior differential ideals on a smooth complex algebraic variety. With these tools we study deformations of several types of differential ideals, leading to several irreducible components of…