Let R be a commutative ring, f ∈ R[X1,⋯,Xk] a multivariate polynomial, and G a finite subgroup of the group of units of R satisfying a certain constraint, which always holds if R is a field. Then, we evaluate Σ f(x1,⋯,xk), where the summation is taken over all pairwise distinct x1,⋯,xk G. In particular, let ps be a power of an odd prime, n a positive integer coprime with p - 1, and a1,⋯,ak integers such that φ(ps) divides a1 + ⋯ + ak and p - 1 does not divide Σi∈Iai for all non-empty proper subsets I ⊆ {1,⋯,k}; then ∑x1a1⋯xkak ≡ φ(ps)/gcd(n,φ(ps))(-1)k-1(k - 1)!modps, where the summation is taken over all pairwise distinct nth residues x1,⋯,xk modulo ps coprime with p.

### Sums of Multivariate Polynomials in Finite Subgroups

#### Abstract

Let R be a commutative ring, f ∈ R[X1,⋯,Xk] a multivariate polynomial, and G a finite subgroup of the group of units of R satisfying a certain constraint, which always holds if R is a field. Then, we evaluate Σ f(x1,⋯,xk), where the summation is taken over all pairwise distinct x1,⋯,xk G. In particular, let ps be a power of an odd prime, n a positive integer coprime with p - 1, and a1,⋯,ak integers such that φ(ps) divides a1 + ⋯ + ak and p - 1 does not divide Σi∈Iai for all non-empty proper subsets I ⊆ {1,⋯,k}; then ∑x1a1⋯xkak ≡ φ(ps)/gcd(n,φ(ps))(-1)k-1(k - 1)!modps, where the summation is taken over all pairwise distinct nth residues x1,⋯,xk modulo ps coprime with p.
##### Scheda breve Scheda completa Scheda completa (DC)
2018
2017
Commutative ring; multivariate polynomial; nth residues; regular element; symmetric sum; Wilson's theorem
Leonetti, P; Marino, A
File in questo prodotto:
File
MultivariatePol.pdf

accesso aperto

Tipologia: Documento in Pre-print
Licenza: Creative commons
Dimensione 170.47 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2142100`