Reconstruction Algorithms for Sums of Affine Powers

TL;DR

This paper investigates sums of affine powers in univariate polynomials, providing structural insights, algorithms for minimal decompositions, and initial steps towards multivariate extensions, highlighting open problems and future directions.

Contribution

It introduces algorithms for minimal affine power decompositions of polynomials and compares the expressive power of related models, advancing understanding of polynomial decomposition.

Findings

Algorithms for minimal affine power decomposition under certain assumptions

Structural comparison of affine powers, Waring, and sparsest shift models

Initial exploration of multivariate cases and open problems

Abstract

In this paper we study sums of powers of affine functions in (mostly) one variable. Although quite simple, this model is a generalization of two well-studied models: Waring decomposition and sparsest shift. For these three models there are natural extensions to several variables, but this paper is mostly focused on univariate polynomials. We present structural results which compare the expressive power of the three models; and we propose algorithms that find the smallest decomposition of f in the first model (sums of affine powers) for an input polynomial f given in dense representation. We also begin a study of the multivariate case. This work could be extended in several directions. In particular, just as for Sparsest Shift and Waring decomposition, one could consider extensions to "supersparse" polynomials and attempt a fuller study of the multi-variate case. We also point out that the basic univariate problem studied in the present paper is far from completely solved: our algorithms all rely on some assumptions for the exponents in an optimal decomposition, and some algorithms also rely on a distinctness assumption for the shifts. It would be very interesting to weaken these assumptions, or even to remove them entirely. Another related and poorly understood issue is that of the bit size of the constants appearing in an optimal decomposition: is it always polynomially related to the bit size of the input polynomial given in dense representation?