Two remarks on polynomials in two variables

Waring’s Problem for Polynomials in Two Variables

We prove that all polynomials in several variables can be decomposed as the sums of kth powers: P (x1, . . . , xn) = Q1(x1, . . . , xn) + · · ·+Qs(x1, . . . , xn), provided that elements of the base field are themselves sums of kth powers. We also give bounds for the number of terms s and the degree of the Qi . We then improve these bounds in the case of two variables polynomials of large degre...

Waring Problem for Polynomials in Two Variables

We prove that all polynomials in several variables can be decomposed as the sum of kth powers: P (x1, . . . , xn) = Q1(x1, . . . , xn) + · · ·+Qs(x1, . . . , xn), provided that elements of the base field are themselves sum of kth powers. We also give bounds for the number of terms s and the degree of the Qi . We then improve these bounds in the case of two variables polynomials to get a decompo...

Remarks on two symmetric polynomials and some matrices

By means of comp lete symmetric polynomials this paper gives a new proof for the Vander-monde determinant formula. Another alternative proof for this formula is obtained via the collocatio n matrices. It also gives a generalized relationship between the Vandermo nde, the Pascal and the Stirling matrices. A new app roach to obtain the explicit inverse of the Vandermo nde matrix is investigated. ...

On Pseudo Hermite Matrix Polynomials of Two Variables

Abstract. The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix p...

Two Addition Theorems on Polynomials of Prime Variables