Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well known.
We consider a class $\mathcal{F}_{q,n}$ of polynomials, which we obtain by modifying linear permutations at $n$ points. The study of the spectrum of $\mathcal{F}_{q,n}$ enables us to obtain a simple description of polynomials $F \in \mathcal{F}_{q,n}$ with prescribed $V_F$, especially those avoiding a given set, like cosets of subgroups of the multiplicative group $\mathbb{F}_q^*$. The value set count for such $F$ can also be determined. This yields polynomials with evenly distributed values, which have small maximum count.
A Note on a Nearly Uniform Partition into Common Independent Sets of Two Matroids
本说明是对K. Takazawa和Y. Yokoi（A generalized-polymatoid approach to disjoint common independent sets in two matroids，Discrete Mathematics（2019））.对于给定的$E$上的两个拟阵，在与他们论文中相同的假设下，保证存在$E$到两个拟阵的$k$公共独立集的划分，我们证明了存在$E$到$k$公共独立集的几乎一致的划分$\mathcal{P}$，其中$\mathcal{P}$中任何两个集合的基数之差最多为1。
