Preliminaries

Emil Grosswald
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Emil Grosswald: Temple University, College of Liberal Arts

Chapter Chapter 1 in Representations of Integers as Sums of Squares, 1985, pp 5-12 from Springer

Abstract: Abstract In this book, when we speak of a quadratic form,* we mean a rational, integral quadratic form, unless the contrary is stated explicitly. Given a quadratic form Q, let N Q be the set of values of Q where x ∈ ℤ k (i.e., x i ∈ ℤ for i = 1, 2, ..., k); clearly, N Q ⊂ ℤ. If $$ Q = \sum\nolimits_{i = 1}^k {x_i^2} $$ , we denote N Q by N k . The main problems that we shall study can now be formulated as follows: (a) Given a quadratic form Q, determine N Q . (b) Given Q and n ∈ N Q , determine the number of representations of n by Q, i.e., the number of vectors x ∈ ℤk for which Q(x) = n.

Keywords: Quadratic Form; Nontrivial Solution; Theta Function; Diophantine Equation; Great Common Divisor (search for similar items in EconPapers)
Date: 1985
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DOI: 10.1007/978-1-4613-8566-0_2

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