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## Publications

Publications (8)

Text
For any fixed field K∈{Q2,Q3,Q5,…}, we prove that all univariate polynomials f with exactly 3 (resp. 2) monomial terms, degree d, and all coefficients in {±1,…,±H}, can be solved over K within deterministic time log4+o(1)(dH)log3d (resp. log2+o(1)(dH)) in the classical Turing model: Our underlying algorithm correctly counts the number of ro...

For any fixed field $K\!\in\!\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5, \ldots\}$, we prove that all polynomials $f\!\in\!\mathbb{Z}[x]$ with exactly $3$ (resp. $2$) monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ within deterministic time $\log^{7+o(1)}(dH)$ (resp. $\log^{2+o(1)}(dH)$) in t...

Let $k,p\in \mathbb{N}$ with $p$ prime and let $f\in\mathbb{Z}[x_1,x_2]$ be a bivariate polynomial with degree $d$ and all coefficients of absolute value at most $p^k$. Suppose also that $f$ is variable separated, i.e., $f=g_1+g_2$ for $g_i\in\mathbb{Z}[x_i]$. We give the first algorithm, with complexity sub-linear in $p$, to count the number of ro...

We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$, we prove that any polynomial $f\in\mathbb{Z}[x_1]$ with exactly $3$ monomial terms, degree $d$, and all coeffi...

Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the number of roots of $f$ in $\mathbb{Z}/\!\left(p^k\right)$ within time $d^3(k\log p)^{2+o(1)}$. (We in fact prove a...

Thanks to earlier work of Koiran, it is known that the truth of the Generalized Riemann Hypothesis (GRH) implies that the dimension of algebraic sets over the complex numbers can be determined within the polynomial-hierarchy. The truth of GRH thus provides a direct connection between a concrete algebraic geometry problem and the P vs.NP Problem, in...