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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Jacobi's bound for first order difference equations


Author: Barbara A. Lando
Journal: Proc. Amer. Math. Soc. 32 (1972), 8-12
MSC: Primary 12.80
DOI: https://doi.org/10.1090/S0002-9939-1972-0289474-X
MathSciNet review: 0289474
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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).


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DOI: https://doi.org/10.1090/S0002-9939-1972-0289474-X
Keywords: Difference polynomial, Jacobi bound, difference kernel, order of an irreducible difference variety
Article copyright: © Copyright 1972 American Mathematical Society