New Bounds on Quotient Polynomials with Applications to Exact Divisibility and Divisibility Testing of Sparse Polynomials
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A sparse polynomial (also called a lacunary polynomial) is a polynomial that has relatively few terms compared to its degree. The sparse-representation of a polynomial represents the polynomial as a list of its non-zero terms (coefficient-degree pairs). In particular, the degree of a sparse polynomial can be exponential in the sparse-representation size. We prove that for monic polynomials f, g ∈ℂ[x] such that g divides f, the ℓ_2-norm of the quotient polynomial f/g is bounded by ‖ f ‖_1 ·Õ(‖g‖_0^3deg^2 f)^‖g‖_0 - 1. This improves upon the exponential (in deg f) bounds for general polynomials and implies that the trivial long division algorithm runs in time quasi-linear in the input size and number of terms of the quotient polynomial f/g, thus solving a long-standing problem on exact divisibility of sparse polynomials. We also study the problem of bounding the number of terms of f/g in some special cases. When f, g ∈ℤ[x] and g is a cyclotomic-free (i.e., it has no cyclotomic factors) trinomial, we prove that ‖f/g‖_0 ≤ O(‖f‖_0 size(f)^2 ·log^6deg g). When g is a binomial with g(± 1) ≠ 0, we prove that the sparsity is at most O(‖f‖_0 ( log‖f‖_0 + log‖f‖_∞)). Both upper bounds are polynomial in the input-size. We leverage these results and give a polynomial time algorithm for deciding whether a cyclotomic-free trinomial divides a sparse polynomial over the integers. As our last result, we present a polynomial time algorithm for testing divisibility by pentanomials over small finite fields when deg f = Õ(deg g).
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