On the algebraic dependence of holonomic functions

关于完整序列的有理递归

On Rational Recursion for Holonomic Sequences

本文证明离散时间有理动力系统分量满足拟线性差分方程的猜想对完整序列成立，并给出两种在效率和输出阶数极小性上不同的转换算法。

On determinantal ideals and algebraic dependence

Let $X$ be a matrix with entries in a polynomial ring over an algebraically closed field $K$. We prove that, if the entries of $X$ outside some $(t \times t)$-submatrix are algebraically dependent over $K$, the arithmetical rank of the ideal $I_t(X)$ of $t$-minors of $X$ drops at least by one with respect to the generic case; under suitable assumptions, it drops at least by $k$ if $X$ has $k$ zero entries. This upper bound turns out to be sharp if $\mathrm{char}\, K=0$, since it then coincides with the lower bound provided by the local cohomological dimension.