A New Bound on Cofactors of Sparse Polynomials
For researchers in computational algebra, this provides the first polynomial bound on cofactor size and quasi-linear time algorithm for exact sparse polynomial division, a previously open problem.
The paper proves a new bound on the ℓ2-norm of cofactors in sparse polynomial division, improving from exponential to polynomial in degree, and shows that exact division runs in quasi-linear time, resolving a long-standing open problem.
We prove that for polynomials $f, g, h \in \mathbb{Z}[x]$ satisfying $f = gh$ and $f(0) \neq 0$, the $\ell_2$-norm of the cofactor $h$ is bounded by $\|h\|_2 \leq \|f\|_1 \cdot\left( \widetilde{O}\left(\|g\|_0^3 \frac{\text{deg }{(f)}^2}{\sqrt{\text{deg }{(g)}}}\right)\right)^{\|g\|_0 - 1}$, where $\|g\|_0$ is the number of nonzero coefficients of $g$ (its sparsity). We also obtain similar results for polynomials over $\mathbb{C}$. This result significantly improves upon previously known exponential bounds (in $\text{deg }{(f)}$) for general polynomials. It further implies that, under exact division, the polynomial division algorithm runs in quasi-linear time with respect to the input size and the number of terms in the quotient $h$. This resolves a long-standing open problem concerning the exact divisibility of sparse polynomials. In particular, our result demonstrates a quadratic separation between the runtime (and representation size) of exact and non-exact divisibility by sparse polynomials. Notably, prior to our work, it was not even known whether the representation size of the quotient polynomial could be bounded by a sub-quadratic function of its number of terms, specifically of $\text{deg }{(f)}$.