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Sparsity bound gives poly-time deterministic exact root for sparse powers
When total degree is bounded, the base of an exact e-th power has at most s to a power linear in D terms, so the root can be recovered deter
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We study the problem of deterministically computing the exact root of a sparse polynomial in the multivariate setting. Let $f \in \F[x_1,\ldots,x_n]$ be a nonzero polynomial that is an exact $e$-th power, say $f = g^e$. Suppose $f$ is $s$-sparse, has an individual degree of at most $d$, and a total degree of $D = \tdeg(f)$. We prove a sparsity bound on the base polynomial $g$: \[ \|g\|_0 \le s^{D(2d+2)/e + 1}. \] Based on this bound, we develop a deterministic algorithm that computes the base $g$. %
In contrast to the general deterministic factorization algorithm of Bhargava, Saraf, and Volkovich \cite{BhargavaSarafVolkovich2020}, which achieves only a quasi-polynomial dependence on the input parameters, our algorithm is \emph{polynomial-time} in the setting where the total degree $D$ is bounded. Specifically, the overall complexity is \[ \mathrm{poly}\left(s^{O(Dd)}, n, d, D\right) + s\cdot R(e), \] %
where $R(e)$ denotes the cost of constructing a single $e$-th root of a scalar in the base field $\F$, and, when $\operatorname{char}(\F)\mid e$, the cost of computing a single Frobenius root of a scalar. %
This term is field-dependent, and over finite fields, $\mathbb{Q}$, or number fields with a suitable representation, it is absorbed into the polynomial complexity bound. %
Within the bounded total-degree regime, this yields a deterministic polynomial-time algorithm for exact-root computation.
