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Counting k-tuples of positive integers such that the values of several polynomials of k variables are relatively prime
Abstract
Let F=(f1(x1,…,xk),…, fm(x1,…,xk)) be a system of nonconstant polynomials of k variables with integer coefficients and let gcdF(x1,…,xk)= gcd(f1(x1,…,xk),…, fm(x1,…,xk)). We obtain an unconditional asymptotic formula for the sum ∑1≤ x1,…,xk≤ x h(gcdF(x1,…,xk)), where F is a system of m≥ 2 polynomials of k≥ 2 variables subject to certain general properties, and h is a bounded strongly multiplicative function. In particular, we deduce an asymptotic formula with error term concerning the density of k-tuples of positive integers (x1,…,xk) such that gcdF(x1,…,xk)=1, given by the Ekedahl-Poonen formula.
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BibTeX
@misc{toth2026counting,
title = {{Counting $k$-tuples of positive integers such that the values of several polynomials of $k$ variables are relatively prime}},
author = {László Tóth},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28284},
eprint = {2609.28284},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28284},
}
RIS
TY - UNPB
TI - Counting k-tuples of positive integers such that the values of several polynomials of k variables are relatively prime
AU - László Tóth
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28284
UR - https://arxiv.org/abs/2609.28284
AB - Let F=(f_1(x_1,…,x_k),…, f_m(x_1,…,x_k)) be a system of nonconstant polynomials of k variables with integer coefficients and let gcd_F(x_1,…,x_k)= gcd(f_1(x_1,…,x_k),…, f_m(x_1,…,x_k)). We obtain an unconditional asymptotic formula for the sum ∑_1≤ x_1,…,x_k≤ x h(gcd_F(x_1,…,x_k)), where F is a system of m≥ 2 polynomials of k≥ 2 variables subject to certain general properties, and h is a bounded strongly multiplicative function. In particular, we deduce an asymptotic formula with error term concerning the density of k-tuples of positive integers (x_1,…,x_k) such that gcd_F(x_1,…,x_k)=1, given by the Ekedahl-Poonen formula.
ER -