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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?
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be written as a s
In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial
In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for
We study the problem of recovering a collection of $n$ numbers from the evaluation of $m$ power sums. This yields a system of polynomial equations, which can be underconstrained ($m < n$), square ($m = n$), or overconstrained ($m > n$). Fibers and im
We show that the diophantine equation $n^ell+(n+1)^ell + ...+ (n+k)^ell=(n+k+1)^ell+ ...+ (n+2k)^ell$ has no solutions in positive integers $k,n ge 1$ for all $ell ge 3$.