🤖 AI Summary
This work addresses the exact divisibility problem for sparse polynomials, aiming to overcome the exponential dependence of classical algorithms on the degree deg(f) of the dividend. We establish the first quasi-polynomial ℓ₂-norm upper bound on the quotient—depending solely on the sparsity ‖g‖₀ of the divisor g and deg(f)—thereby achieving the first subexponential norm estimate for quotients. We further prove that exact division can be performed in time quasi-linear in both the input size and the number of terms in the quotient. Our analysis reveals a fundamental quadratic separation between exact and inexact divisibility, both in computational complexity and in the representation size of the quotient. Finally, we resolve a long-standing open question: the size of the quotient in sparse polynomial divisibility *can* indeed be bounded by a subquadratic function of its number of terms—settling the conjecture on tight control of quotient magnitude via term count.
📝 Abstract
We prove that for polynomials $f, g, h in mathbb{Z}[x]$ satisfying $f = gh$ and $f(0)
eq 0$, the $ell_2$-norm of the cofactor $h$ is bounded by $|h|_2 leq |f|_1 cdotleft( widetilde{O}left(|g|_0^3 frac{ ext{deg }{(f)}^2}{sqrt{ ext{deg }{(g)}}}
ight)
ight)^{|g|_0 - 1}$, 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 $ ext{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 $ ext{deg }{(f)}$.
