Counting Isolated Roots of Trinomial Systems in the Plane and Beyond

Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a si…

APN trinomials and hexanomials

In this paper we give a new family of APN trinomials of the form $X^{2^k+1} + (\mathsf{tr}^{n}_{m}(X))^{2^k+1}$ on $\mathbb{F}_{2^n}$ where $\mathsf{gcd}(k,n)=1$ and $n = 2m = 4t$, and prove its important properties. The family satisfies for all $n = 4t$ an interesting property of the Kim function which is, up to equivalence, the only known APN function equivalent to a permutation on $\mathbb{F}_{2^{2m}}$. As another contribution of the paper, we consider a family of hexanomials $g_{C,k}$ which was shown to be differentially $2^{\mathsf{gcd}(m,k)}$-uniform by Budaghyan and Carlet (2008) when a…

关于$N=3n$字母单词数量的问题

On the Number of Words of Length $N$ over a Three-Letter Alphabet

本文研究了三字母字母表中$3n$字母单词的数量问题，通过分解为四种情况来计数，并给出了三项式系数的和。

邻域条件下的施赖尔集计数

Counting Schreier Sets Under Neighborhood Conditions

该研究计数满足各类邻域条件的施赖尔集，为前四种条件确定初始计数并证明线性递推，为第五种条件证明含除数计数函数的递推关系。

计数齐性爱因斯坦度量

Counting Homogeneous Einstein Metrics

本文给出紧致齐性空间上孤立齐性爱因斯坦度量数量的显式上界，基于BKK界和组合体积计算，并确认了Böhm--Wang--Ziller有限性猜想的特殊情形，同时用数值代数几何计算了低维全旗流形上的不变度量。

用 Kromatic 对称函数计数诱导子图

Counting induced subgraphs with the Kromatic symmetric function

本文研究 Kromatic 对称函数的图区分能力，猜想其可区分所有图，并证明它能确定若干 4、5 顶点诱导子图及星加孤立点图的数量。