Birch's theorem with shifts

2014/10/28 by Chow, Sam
#11D75 (primary) #11E76 (secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1410.7789

Abstract

Let f₁, ..., f_R be rational forms of degree d ≥ 2 in n > σ+ R(R+1)(d-1)2^d-1 variables, where σ is the dimension of the affine variety cut out by the condition rank(∇ f_k)_k=1^R < R. Assume that f = 0 has a nonsingular real solution, and that the forms (1,...,1) ⋅ ∇ f_k are linearly independent. Let \boldsymbolτ ∈ ℝ^R, let μ be an irrational real number, and let η be a positive real number. We consider the values taken by f(x₁ + μ, ..., x_n + μ) for integers x₁, ..., x_n. We show that these values are dense in ℝ^R, and prove an asymptotic formula for the number of integer solutions x ∈ [-P,P]ⁿ to the system of inequalities |f_k(x₁ + μ, ..., x_n + μ) - τ_k| < η (1 ≤ k≤ R).

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