Finite Basis Property of S-Ideals of Finite Dimensional Forms

V. V. Shchigolev
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V. V. Shchigolev: Moscow State University

A chapter in Formal Power Series and Algebraic Combinatorics, 2000, pp 581-592 from Springer

Abstract: Abstract In the present article we consider finite basis property of some systems of commutative polynomials. In addition, we require sometimes that these systems are invariant under the action of the group GL∞(K). In this article we call such systems S-ideals. In the case of zero characteristic S-ideals are studied in [5]. Let K be a field, X (r) = {x i (j) : i ∈ N, j = 1,..., r} a countable set of variables, and F = K[X (r)] a free commutative algebra over K. Take some abstract vectors e l,..., e r and let W = Fe 1 + ... + Fe r be a free F-module. Fix the following elements $$\mathop {{\text{ }}x}\limits_i^ - = x_i^{\left( 1 \right)}{e_1} + \cdots + x_i^{\left( r \right)}{e_r}$$ in W, which we call generic vectors. Let F 1,..., F q be F-linear maps F i: $${W^{ \times {m_i}}} \to F$$ such that $$F({e_{{j_i}}}, \cdots ,{e_{{j_i}}})$$ ∈K By G denote the subalgebra of F generated by the polynomials $${F_i}({\bar x_{j1}}, \ldots ,{\bar x_{jmi}})$$ . Let us take a monomial $$u = {F_{i1}}({\bar x_{j(1,1)}}, \ldots ,{\bar x_{j(1,{m_{i1}})}}) \ldots {F_{is}}({\bar x_{j(s,1)}}, \cdots ,{\bar x_{j(s,{m_{is)}}}})$$ of the algebra G. The number of occur-rences of i In the sequence $$j(1,1), \cdots ,j(1,{m_{{i_1}}}), \cdots ,j(s,1), \cdots ,j(s,{m_{{i_s}}})$$ is called the degree of u with respect to $${\bar x_i}$$ and is denoted by $${\deg _{\bar xi}}$$ u. Let us expand u to a linear combination of monomials in variables from X (r) and take an arbitrary monomial of the obtained expression $$x_{{k_1}}^{{l_1}} \cdots x_{{k_t}}^{{l_t}}$$ , where $$t = {m_{{i_1}}} + \cdots {m_{{i_s}}}$$ . Then the number of occurrences of i in the sequence l t,..., l t is equal to $${\deg _{\bar xi}}$$ u. This shows that the degree $${\deg _{\bar xi}}$$ u is well defined.

Date: 2000
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DOI: 10.1007/978-3-662-04166-6_56

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