Linearization of multivariate discrete difference operators

TL;DR

This paper unifies approaches to multivariate discrete difference operators, proving that the functional degree of integer-valued maps remains consistent across different difference operator bases, thus advancing the linearization of these operators.

Contribution

It demonstrates the equivalence of functional degrees computed with respect to standard and arbitrary multivariate difference operators, unifying previous methods.

Findings

Functional degree is invariant across difference operator bases.

Linearization of multivariate difference operators is achieved.

Unification of two approaches to difference operators.

Abstract

In 2023 in (3), Uwe finds the explicit form of the map which is which is settled in ZN of finite functional degree and14 discusses how to compute its usual degree w.r.t to the derivative in the linear form, i.e. the product of ones formed by15 its orthogonal basis, and also introduces the notion of functional degree. It is the linear combination of the product of16 binomials. And this makes a big progress in the development of integer-valued functions. And this inspires the author17 to discuss this form in the separate form, i.e. the triple of finite set. In (18), Hrycaj discusses these operators in more18 abstract form. In this paper, we unifies two approaches. Seriously, we prove that the functional degree of integer-valued19 maps on integers which is computed with respect to its difference operators associated with its standard basis is exactly20 the same with the ones computed with respect to arbitrary mix discrete difference operators and we call it as multivariate21 difference operators. Since the functional degree agrees with the usual degree of integer-valued maps, we actually achieved22 the linearization of multivariate difference operators