By Bernd Sturmfels
This booklet is either an easy-to-read textbook for invariant concept and a demanding examine monograph that introduces a brand new method of the algorithmic facet of invariant concept. scholars will locate the publication a simple creation to this "classical and new" zone of arithmetic. Researchers in arithmetic, symbolic computation, and desktop technology gets entry to analyze rules, tricks for purposes, outlines and information of algorithms, examples and difficulties.
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Extra resources for Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation)
Clearly, (b) implies (a). To prove the converse, suppose that Â1 ; : : : ; Ân is a regular sequence in R and that 1 ; : : : ; n is any h. s. o. p. We need to show that 1 ; : : : ; n is a regular sequence. R/. RC / a parameter. In other words, Â is not a zero-divisor, and R is a finitely generated CŒ -module. RC / such that u D 0 in R. u/ D fv 2 R j v u D 0g. u/ is zero-dimensional. u/ for some m 2 N. This means that Â m is a zero-divisor and hence not regular. 2, because Â was assumed to be regular.
5), every ideal in the polynomial ring CŒx is finitely generated. Hence there exist finitely many homogeneous invariants I1 ; I2 ; : : : ; Im such that I D hI1 ; I2 ; : : : ; Im i. We shall now prove that all homogeneous invariants I 2 CŒx can actually be written as polynomial functions in I1 ; I2 ; : : : ; Im . Suppose the contrary, and let I be a homogeneous elementPof minimum degree in CŒx n CŒI1 ; I2 ; : : : ; Im . I /. 2. I /. From the minimality assumption on I we get fj 2 CŒI1 ; : : : ; Im and therefore I 2 CŒI1 ; : : : ; Im , which is a contradiction to our assumption.
Therefore the minimum distance d between code words equals the smallest weight of any nonzero code word. The weight enumerator of an Œn; k; d code C is a bivariate polynomial which tells the number of code words of each weight. x1 ; x2 / WD n P ai x1n i x2i : iD0 Notice that WC is a homogeneous polynomial of degree n. The weight enumerator immediately gives the minimum distance d of C. For C always contains the zero code word, giving the leading monomial x1n of WC , and the next nonzero monomial is ad x1n d x2d .
Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation) by Bernd Sturmfels