In this paper we explore the interrelationship between the theory of polynomial ideals and certain branches of analysis including multivariate approximation theory and linear partial differential equations. Specifically, we use the perturbation technique from analysis to study the codimension of a multivariate polynomial ideal. Let K be a field. In this paper, K will often be the field C of com...

We give upper bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over the field of complex numbers. Let x be a set of n differential variables, f a finite family of differential polynomials in the ring C{x} and f ∈ C{x} another polynomial which vanishes at every solution of the differential equation system f = 0 in any differentially ...

We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor. In particular, we show that the module of syzygies of polynomials f1, . . . , fn ∈ R[X1, . . . , XN ] with coefficients in a Prüfer domain R can be generated by elements whose degrees are bounded by a number only depending on N ...

In Mn(k), k an algebraically closed field, we call a matrix l-regular if each eigenspace is at most l-dimensional. We prove that the variety of commuting pairs in the centralizer of a 2-regular matrix is the direct product of various affine spaces and various determinantal varieties Zl,m obtained from matrices over truncated polynomial rings. We prove that these varieties Zl,m are irreducible, ...

We give an effective solution of the conjugacy problem for two by two matrices over the polynomial ring in one variable over a finite field.

We prove that the F-signature of an affine semigroup ring of positive characteristic is always a rational number, and describe a method for computing this number. We use this method to determine the F-signature of Segre products of polynomial rings, and of Veronese subrings of polynomial rings. Our technique involves expressing the F-signature of an affine semigroup ring as the difference of th...

this paper describes the impact of the distance and the size of the corona ring, on the magnitude anddistribution of electrical potential across the polymer insulators. the procedure is based on finiteelement method numerical analysis and stochastic optimization algorithm of differential evolution(de). the optimal selection of corona ring has a significant impact on the control of potential and...