p-adic quotient sets: Cubic forms

2021/10/11 by Antony, Deepa, Barman, Rupam
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2110.05420

Abstract

For A⊆ \1, 2, …\, we consider R(A)=\a/b: a, b∈ A\. It is an open problem to study the denseness of R(A) in the p-adic numbers when A is the set of nonzero values assumed by a cubic form. We study this problem for the cubic forms ax³+by³, where a and b are integers. We also prove that if A is the set of nonzero values assumed by a non-degenerate, integral and primitive cubic form with more than 9 variables, then R(A) is dense in ℚ_p.

Related

• p-Adic quotient sets: diagonal forms
• On p-adic denseness of quotients of values of integral forms
• A note on p-adic denseness of quotients of values of quadratic forms
• Quartic Forms in Many Variables
• p-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials