Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

Let $\mathbb{F}[X]$ be the polynomial ring over the variables $X={x1,x2, \ldots, xn}$. An ideal $I=\langle p1(x1), \ldots, pn(xn)\rangle$ generated by univariate polynomials ${pi(xi)}{i=1}^n$ is a \emph{univariate ideal}. We study the ideal membership problem for the univariate ideals and show the following results. \item Let $f(X)\in\mathbb{F}[\ell1, \ldots, \ellr]$ be a (low rank) polynomial given by an arithmetic circuit where $\elli : 1\leq i\leq r$ are linear forms, and $I=\langle p1(x1), \ldots, pn(xn)\rangle$ be a univariate ideal. Given $\vec{\alpha}\in {\mathbb{F}}^n$, the (unique) remainder $f(X) \pmod I$ can be evaluated at $\vec{\alpha}$ in deterministic time $d^{O(r)}\cdot poly(n)$, where $d=\max{\deg(f),\deg(p1)\ldots,\deg(pn)}$. This yields an $n^{O(r)}$ algorithm for minimum vertex cover in graphs with rank-$r$ adjacency matrices. It also yields an $n^{O(r)}$ algorithm for evaluating the permanent of a $n\times n$ matrix of rank $r$, over any field $\mathbb{F}$. Over $\mathbb{Q}$, an algorithm of similar run time for low rank permanent is due to Barvinok[Bar96] via a different technique. \item Let $f(X)\in\mathbb{F}[X]$ be given by an arithmetic circuit of degree $k$ ($k$ treated as fixed parameter) and $I=\langle p1(x1), \ldots, pn(xn)\rangle$. We show in the special case when $I=\langle x1^{e_1}, \ldots, xn^{en}\rangle$, we obtain a randomized $O^*(4.08k)$ algorithm that uses $poly(n,k)$ space. \item Given $f(X)\in\mathbb{F}[X]$ by an arithmetic circuit and $I=\langle p1(x1), \ldots, pk(xk) \rangle$, membership testing is $W[1]$-hard, parameterized by $k$. The problem is $MINI[1]$-hard in the special case when $I=\langle x1^{e1}, \ldots, xk^{ek}\rangle$.