In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a represen
tation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order $Omega$($sqrt$ d), and were obtained from arguments based on Wronskian determinants and shifted derivatives. We obtain this improvement thanks to a new lower bound method based on Birkhoff interpolation (also known as lacunary polynomial interpolation).
We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $ell^n$ where $ell$ is the lacunary size of the input polynomial and $n$ its number o
f variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial in $ell$ and $n$. Over other fields of characteristic zero and finite fields of large characteristic, our algorithms compute the multilinear factors having at least three monomials of multivariate polynomials. Lower bounds are provided to explain the limitations of our algorithm. As a by-product, we also design polynomial-time deterministic polynomial identity tests for families of polynomials which were not known to admit any. Our results are based on so-called Gap Theorem which reduce high-degree factorization to repeated low-degree factorizations. While previous algorithms used Gap Theorems expressed in terms of the heights of the coefficients, our Gap Theorems only depend on the exponents of the polynomials. This makes our algorithms more elementary and general, and faster in most cases.
