Jacobi’s bound for first order difference equations
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- by Barbara A. Lando
- Proc. Amer. Math. Soc. 32 (1972), 8-12
- DOI: https://doi.org/10.1090/S0002-9939-1972-0289474-X
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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).References
- Richard M. Cohn, Difference algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1965. MR 0205987
- Barbara A. Lando, Jacobi’s bound for the order of systems of first order differential equations, Trans. Amer. Math. Soc. 152 (1970), 119–135. MR 279079, DOI 10.1090/S0002-9947-1970-0279079-1
- J. F. Ritt, Jacobi’s problem on the order of a system of differential equations, Ann. of Math. (2) 36 (1935), no. 2, 303–312. MR 1503224, DOI 10.2307/1968572
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- 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