Reconstruction algorithms for sums of affine powers

Let F be any characteristic zero field and let f∈F[x] be a univariate polynomial. A sum of affine powers is an expression of the formf(x)=∑i=1sαi(x−ai)ei. Although quite simple, this model is a generalization of two well-studied models: Waring decomposition and Sparsest Shift. 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. 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 study the multivariate version of the problem. 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 ei in a decomposition of f, and some algorithms also rely on a distinctness assumption for the shifts ai. 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 ai,αi in an optimal decomposition: is it always polynomially related to the bit size of the input polynomial f given in dense representation?