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Let R be a commutative ring with 1, let R[X 1 , . . . , Xn] be the polynomial ring in X 1 , . . . , Xn over R, and let G be an arbitrary group of permutations of {X 1 , . . . , Xn}. This note presents a detailed analysis and a constructive combinatorial description of SAGBI bases for the R-algebra of G-invariant polynomials. Our main result is a ground ring independent characterization of all rings of polynomial invariants of permutation groups G having a finite SAGBI basis. Definition 3.1. Let B be a SAGBI (Subalgebra Analogue to Gröbner Basis for Ideals) basis of R[X 1 , . . . , X n ] G , i.e. every head term in R[X 1 , . . . , X n ] G can be expressed as a product of head terms in B. Then B is called simple, if all elements of B are non-constant G-invariant orbits. Lemma 3.2. Let B 1 , B 2 be simple SAGBI bases of R[X 1 , . . . , X n ] G . Then B = B 1 ∩ B 2 is a simple SAGBI basis of R[X 1 , . . . , X n ] G .Proof. The lemma follows by induction on the total degree d of the G-invariant orbits in B 1 and B 2 : First, we see that orbit G (X i ) ∈ B 1 and orbit G (X i ) ∈ B 2 for any variable X i , 1 ≤ i ≤ n, and so orbit G (X i ) ∈ B 1 ∩ B 2 . Next, we assume that the statement is true for all orbit G (t) with a total degree less than d, i.e. orbit G (t) is either an element

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We present a parallel Knuth-Bendiz completion algorithm where the inner loop, deriving the consequences of adding a new rule to the system, is multi-threaded. The selection of the best new rule in the outer loop, and hence the completion strategy, is exactly the same as for the sequential algorithm. Our implementation, which is within the PARSA C-2 parallel symbolic computation system, exhibits good parallel speed-UPS on a standard multi-processor workstation.

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