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Muxi Li

Publications and source records attributed to Muxi Li.

7 recordsLinked to original sources
arXiv2026

Frobenius lifts and point counting for smooth curves

We describe an algorithm for computing the zeta function of a proper, smooth curve over a finite field $k$ of characteristic $p$, when the curve is given together with some auxiliary data, including a lift $C$ to the valuation ring in a finite extension of $\Q_p$. The algorithm is denominator-free if the ramification is at most $p$. Our method computes the matrix of the action of a semilinear Frobenius on the first de Rham cohomology group of the curve by means of Poincaré duality, using cup products that can be computed from local expansions of a globally defined lift of Frobenius. Its complexity is softly cubic in the field degree for (general) smooth, planar curve, for which we work out our general estimates in more detail. We make explicit how to compute a suitable basis of the first de Rham cohomology group of $C$, base on 1-forms with `locally integrable polar parts', in both the general case and when the curve is smooth planar. We show the crystalline Frobenius preserves the first de Rham cohomology group of $C$ if the ramification is at most $p$, improving upon known results. In an appendix we prove a well-known formula for the cup product, and a formula by Serre, on the first de Rham cohomology group for a curve in characteristic zero, for which no reference seems to exist.

math.AG↗
arXiv2026

Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles

For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, Δ(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $Δ(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $Δ(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $Δ(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $Δ(9)$ remains open.

math.CO↗
arXiv2026

A Weak Solution of Inverse Treblecross

We give a weak solution for an impartial game we called ``Inverse Treblecross''. We have determined which of its starting positions are $\mathscr{P}$-position and give a reasonable strategy.

math.CO↗
arXiv2020

Characterization of Beauville's algebraic numbers via Hodge theory

We provide a Hodge theoretical characterization of the set of algebraic numbers which arises from the complete list, due to A. Beauville, of semistable families of elliptic curves over $\mathbb{P}^1$ with four singular fibers. Our technical innovation is the analysis of the periodicity of the uniformizing Higgs bundle attached to $\mathbb{P}^1$ minus four points over the field of complex numbers.

math.AG↗
arXiv2018

Integral Regulators for Higher Chow Complexes

Building on Kerr, Lewis and Mueller-Stach's work on the rational regulator, we prove the existence of an integral regulator on higher Chow complexes and give an explicit expression. This puts firm ground under some earlier results and speculations on the torsion in higher cycle groups by K-L-M, Petras, and Kerr-Yang.

math.AG↗
arXiv2017

A Note on Linear Higher Chow Groups

We give a counterexample to the proof in the literature [K-Theory 25 (2002), 215-231] of the existence of linear representatives of higher Chow groups of number fields.

math.AG↗