New Bounds on Quotient Polynomials with Applications to Exact Division and Divisibility Testing of Sparse Polynomials

We prove that for monic polynomials f, g ϵ C[x] such that g divides f, the ℓ₂-norm of the quotient f/g is bounded by ||f||₁ O(||{g}||₀³deg²f)^||{g}||0-1, improving upon the previously known exponential (in deg (f)) bounds for general polynomials. This result implies that the trivial long division algorithm runs in quasi-linear time relative to the input size and number of terms of the quotient, thus solving a long-standing problem. We also bound 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·log⁶deg 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. Leveraging these results, we provide a polynomial-time algorithm for deciding whether a cyclotomic-free trinomial divides a sparse polynomial over the integers.