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Jacobi's bound for the order of systems of first order differential equations


Author: Barbara A. Lando
Journal: Trans. Amer. Math. Soc. 152 (1970), 119-135
MSC: Primary 12.80
DOI: https://doi.org/10.1090/S0002-9947-1970-0279079-1
MathSciNet review: 0279079
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Abstract: Let $ {A_1}, \ldots ,{A_n}$ be a system of differential polynomials in the differential indeterminates $ {y^{(1)}}, \ldots ,{y^{(n)}}$, and let $ \mathcal{M}$ be an irreducible component of the differential variety $ \mathcal{M}({A_1}, \ldots ,{A_n})$. If $ \dim \mathcal{M} = 0$, there arises the question of securing an upper bound for the order of $ \mathcal{M}$ in terms of the orders $ {r_{ij}}$ of the polynomials $ {A_i}$ in $ {y^{(j)}}$. It has been conjectured that the Jacobi number

$\displaystyle J = J({r_{ij}}) = \max \{\sum\limits_{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text{ is a permutation of 1,}} \ldots ,n \}$

provides such a bound. In this paper $ J$ is obtained as a bound for systems consisting of first order polynomials. Differential kernels are employed in securing the bound, with the theory of kernels obtained in a manner analogous to that of difference kernels as given by R. M. Cohn.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0279079-1
Keywords: Differential polynomial, Jacobi bound, differential kernel, specialization, dimension, order of an irreducible differential variety
Article copyright: © Copyright 1970 American Mathematical Society

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