Reconstruction Algorithms for Sums of Affine Powers

Abstract

A sum of affine powers is an expression of the formAlthough 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 multivariate 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 e i in a decomposition of f , and some algorithms also rely on a distinctness assumption for the shifts a i . 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 a i , α i in an optimal decomposition: is it always polynomially related to the bit size of the input polynomial f given in dense representation?

On the linear independence of shifted powers

“…Case k = s: we obtain l ≥ s(s + 1)/2. This is similar to the SDE coming from the factorized Wronskian exhibited in the first version of [7].…”

“…The name SDE is a reference to Neeraj Kayal's lower bound method of shifted partial derivatives [16]. We have already used shifted differential equations in the design of efficient algorithms for shifted powers [7]. The main novelty in the present paper is a technical lemma (see Proposition 2.2 and Corollary 2.5) which helps to deal with families where some nodes a i are repeated, i.e., occur in several shifted powers of the family.…”

“…We generalize this observation to several shifted powers in Proposition 2.2 below. Shifted differential equations have already been used in [10] to obtain lower bounds and in [7] to obtain efficient algorithms. In these two papers the authors worked with a slightly less general definition of SDEs: they considered SDEs of order k, shift l and the degrees of the P i 's were all of the form deg P i ≤ i + l. This corresponds now to the choice of parameters (k + 1, k, l) in Definition 2.1, i.e., with t = k + 1.…”

“…The method of shifted differential equations is quite robust, and this allows us to obtain a key ingredient similar to that of [7]: for a family of s shifted powers, there exists a "small" SDE satisfied by all the terms. More precisely:…”

“…where e i ∈ N and the a i belong to a field K of characteristic 0. An element of F will be called a shifted power (polynomials of this form are also called affine powers in [7]). We always assume that (a i , e i ) = (a j , e j ) for i = j, i.e.…”