Jacobi’s bound for first order difference equations

Abstract:

Let ${A_1}, \cdots ,{A_n}$ be a system of difference polynomials in ${y^{(1)}}, \cdots ,{y^{(n)}}$, and let $\mathcal {M}$ be an irreducible component of the difference variety $\mathcal {M}({A_1}, \cdots ,{A_n})$. If ${r_{ij}}$ is the order of ${A_i}$ in ${y^{(j)}}$, the Jacobi number J of the system is defined to be $\max \{ \sum _{i = 1}^n{r_{i{j_i}}}:{j_1}, \cdots ,{j_n}$ is a permutation of $1, \cdots ,n\}$. In this paper it is shown for first order systems that if $\dim \mathcal {M} = 0$, then E ord $\mathcal {M} \leqq J$. The methods used are analogous to those used to obtain the corresponding result for differential equations (given in a recent paper by the author).