arXiv subjects

Weixun Deng

Publications and source records attributed to Weixun Deng.

3 recordsLinked to original sources

arXiv2025

Optimal Bounds for the Number of Pieces of Near-Circuit Hypersurfaces

Suppose $f$ is a polynomial in $n$ variables with real coefficients, exactly $n+k$ monomial terms, and Newton polytope of positive volume. Estimating the number of connected components of the positive zero set of $f$ is a fundamental problem in real algebraic geometry, with applications in computational complexity and topology. We prove that the number of connected components is at most $3$ when $k\!=\!3$, settling an open question from Fewnomial Theory. Our results also extend to exponential sums with real exponents. A key contribution here is a deeper analysis of the underlying $\mathcal{A}$-discriminant curves, which should be of use for other quantitative geometric problems.

math.AG↗

arXiv2024

Viro's patchworking and the signed reduced A-discriminant

Computing the isotopy type of a hypersurface, defined as the positive real zero set of a multivariate polynomial, is a challenging problem in real algebraic geometry. We focus on the case where the defining polynomial has combinatorially restricted exponent vectors and fixed coefficient signs, enabling faster computation of the isotopy type. In particular, Viro's patchworking provides a polyhedral complex that has the same isotopy type as the hypersurface, for certain choices of the coefficients. So we present properties of the signed support, focussing mainly on the case of n-variate (n+3)-nomials, that ensure all possible isotopy types can be obtained via patchworking. To prove this, we study the signed reduced A-discriminant and show that it has a simple structure if the signed support satisfies some combinatorial conditions.

math.AG↗

arXiv2022

Trinomials and Deterministic Complexity Limits for Real Solving

We detail an algorithm that -- for all but a $\frac{1}{\Omega(\log(dH))}$ fraction of $f\in\mathbb{Z}[x]$ with exactly $3$ monomial terms, degree $d$, and all coefficients in $\{-H,\ldots, H\}$ -- produces an approximate root (in the sense of Smale) for each real root of $f$ in deterministic time $\log^{4+o(1)}(dH)$ in the classical Turing model. (Each approximate root is a rational with logarithmic height $O(\log(dH))$.) The best previous deterministic bit complexity bounds were exponential in $\log d$. We then relate this to Koiran's Trinomial Sign Problem (2017): Decide the sign of a degree $d$ trinomial $f\in\mathbb{Z}[x]$ with coefficients in $\{-H,\ldots,H\}$, at a point $r\!\in\!\mathbb{Q}$ of logarithmic height $\log H$, in (deterministic) time $\log^{O(1)}(dH)$. We show that Koiran's Trinomial Sign Problem admits a positive solution, at least for a fraction $1-\frac{1}{\Omega(\log(dH))}$ of the inputs $(f,r)$.

math.AG↗