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Computing Bases for Rings of Permutation-invariant Polynomials

✍ Scribed by Manfred Göbel

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
225 KB
Volume
19
Category
Article
ISSN
0747-7171

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✦ Synopsis

Let (R) be a commutative ring with 1 , let (R\left[X_{1}, \ldots, X_{n}\right]) be the polynomial ring in (X_{1}, \ldots, X_{n}) over (R) and let (G) be an arbitrary group of permutations of (\left{X_{1}, \ldots, X_{n}\right}). The paper presents an algorithm for computing a small finite basis (B) of the (R)-algebra of (G)-invariant polynomials and a polynomial representation of an arbitrary (G)-invariant polynomial in (R\left[X_{1}, \ldots, X_{n}\right]) as a polynomial in the polynomials of the finite basis (B). The algorithm works independently of the ground ring (R), and the basis (B) contains only polynomials of total degree (\leq \max {n, n(n-1) / 2}), independent of the size of the permutation group (G).

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