On the representation of integers by quadratic forms

Abstract

Abstract. Let n 4, and let Q ∈ Z[X1, . . . , Xn] be a non-singular quadratic form. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q = 0, and when Q is positive definite we provide improved upper bounds for the least positive integer k for which the equation Q = k is insoluble in integers, despite being soluble modulo every prime power.

Quadratic Forms Representing All Odd Positive Integers

“…In [5], Browning and Dietmann use the circle method to study integer-matrix quadratic forms Q( x) = x T A x. A pair (Q, k) (consisting of a quaternary quadratic form and a positive integer k) satisfies the strong local solubility condition if for all primes p there is a vector x ∈ Z 4 so that Q( x) ≡ k (mod p 1+2τp ) and p ∤ A x.…”

Density of integer solutions to diagonal quadratic forms

“…In fact our work may be further simplified by appealing to the author's joint work with Dietmann [2], in which uniform estimates for the average order of S q (c) are provided for n 5. The outcome of this investigation is the following result [2, Lemma 7].…”

“…Suppose first that p = 2. Then an application of [2,Lemma 4] We now turn to an upper bound for the factors σ p = D p (n; 0), for odd p | ∆ Q . Recall that σ p = lim k→∞ p −k(n−1) N k (p), N k (p) = #{x (mod p k ) : Q(x) ≡ 0 (mod p k )}.…”

Quantitative results on Diophantine equations in many variables

“…where the constant c n is explicit and depends only on n, has the best possible exponent [Cas55,Cas56]. However, for generic quadratic forms one can do much better [BD08]. Recently, Sardari proved an optimal strong approximation theorem for f − N, where f is a non-degenerate quadratic form and N a sufficiently large integer [Sar15].…”