A minimal Set of Generators for the Ring of multisymmetric Functions
By David Rydh. 2007
25 citations recorded by the source corpus.
The purpose of this article is to give, for any (commutative) ring A, an explicit minimal set of generators for the ring of multisymmetric functions TSAd[x(1),...,x(r)]) = (A[x(1),...,x(r)](circle times)A(d))8(d) A as an A-algebra. In characteristic zero, i.e. when A is a Q-algebra, a minimal set of generators has been known since the 19(th) century. A rather si-nall generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound oil the generators, improving the degree bound previously obtained by Fleischmann. As Gamma(d)(A) (A[x(1),...,x(r)]) = TSAd (A[x(1),...,x(r)]) we also obtain generators for di A vided powers algebras: If B is a finitely generated A-algebra with a given surjection A[x(1), x(2),...,x(r)] --> B then using the corresponding surjection Gamma(d)(A) (A[x(1),...,x(r)]) --> Gamma(d)(A) (B) we get generators for Gamma(d)(A) (B).
Subfields: Algebra and Number Theory, Mathematical Physics
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In Algebra and Number Theory
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Citations in the corpus
- Set2Graph: Learning Graphs From Sets (2020, background): “Since S n − k invariant polynomials with variables x k +1 , . . . , x n are generated by the power-sum multi-symmetric polynomials n i = k +1 with | α | ≤ n − k , see e.g., (Rydh, 2007), we have that each p α ( x k +1 , . . . , ) , for some S n invariant polynomials r α .”
- Provably Powerful Graph Networks (2019, methodology): “We advocate a novel representation of multisets based on Power-sum Multi-symmetric Polynomials (PMP) which are a generalization of the well-known elementary symmetric polynomials.”
- On Universal Equivariant Set Networks (2019, methodology): “Furthermore, every S n invariant polynomial p : R n × k → R can be expressed as a polynomial in the power-sum multi-symmetric polynomials (see (Rydh, 2007) corollary 8.4), i.e., p ( X ) = q ( s 1 ( X ) , . . . , s t ( X )) , (2) These polynomials were recently used to encode multi-sets in Maron et…”