Download e-book for kindle: Algorithms in Invariant Theory (Texts and Monographs in by Bernd Sturmfels

By Bernd Sturmfels

ISBN-10: 3211774165

ISBN-13: 9783211774168

ISBN-10: 3211774173

ISBN-13: 9783211774175

This publication is either an easy-to-read textbook for invariant concept and a not easy examine monograph that introduces a brand new method of the algorithmic aspect of invariant idea. scholars will locate the booklet a simple creation to this "classical and new" quarter of arithmetic. Researchers in arithmetic, symbolic computation, and desktop technology gets entry to investigate rules, tricks for purposes, outlines and info of algorithms, examples and difficulties.

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Extra info for Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation)

Example text

I/ . i/ . Let r be the number of reflections in € and therefore in H . 5 we have j€j D d1 d2 : : : dn D e1 e2 : : : en D jH j, and hence H D €. G The “only-if” part is useful in that it proves that most invariant rings are not polynomial rings. 6 (Twisted symmetric polynomials). C n /, and consider its representation € WD fsign. / W 2 Sn g. We call the elements of the invariant ring CŒx€ twisted symmetric polynomials. Note that a homogeneous polynomial f is twisted symmetric if and only if f B D sign.

4. Reflection groups 49 where deg. j / D ej . Clearly CŒx€  CŒxH , so each Âi is a polynomial function in the ’s. @Âi =@ j / is nonzero. i/ . i/ . Let r be the number of reflections in € and therefore in H . 5 we have j€j D d1 d2 : : : dn D e1 e2 : : : en D jH j, and hence H D €. G The “only-if” part is useful in that it proves that most invariant rings are not polynomial rings. 6 (Twisted symmetric polynomials). C n /, and consider its representation € WD fsign. / W 2 Sn g. We call the elements of the invariant ring CŒx€ twisted symmetric polynomials.

X12 x22 /2 . 11. The weight enumerator of every self-dual binary code is a polynomial function in Â1 and Â2 . 7. We have the representation WC2 D Â14 4 Â2 in terms of fundamental invariants. One of the main applications of Sloane’s approach consisted in proving the nonexistence of certain very good codes. , minimum distance) of the code are expressed in a tentative weight enumerator W , and invariant theory can then be used to show that no such invariant W exists. Exercises (1) Compute the Hilbert series of the ring CŒ 1 ; 2 ; : : : ; n  of symmetric polynomials.

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Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation) by Bernd Sturmfels


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