Continuous Pythagoras numbers for rational quadratic forms

Abstract

We introduce the concept of the continuous Pythagoras number P c ( S) of a subset S of a commutative topological ring to be, roughly, the least number m ≤ ∞ such that the set of sums of squares of elements of S can be represented as sums of m squares of elements of S, by means of m continuous functions. Heilbronn had already shown that P c ( Q ) = 4. Letting L n ( F) be the set of linear n-ary forms over the field F, we show that P c(L n ( R )) = n. We then allow continuously varying nonnegative rational “weights” on the m square summands. If these continuous weight functions and the continuous functions giving the coefficients of the m linear forms, are required to be Q -rational functions of the coefficients of the given positive semidefinite quadratic forms, then we show that P c(L 1 ( R )) = 1 and P c(L n ( R )) = ∞ for n > 1. However, if only the product of the weight functions and the coefficient functions is required to be continuous, then n ≤ P c ( L n ( R )) < [ n! e] (where e is the base of the natural logarithms) and 2 < P c ( L 2( R )); we conjecture that n < P c ( L n ( R )) also for n > 2. On the other hand, if these weight functions and coefficient functions are required only to be rational in the weaker sense of taking rational values at rational arguments, then P c ( L 2( Q )) = 2, and we conjecture that P c ( L n ( Q )) = n also for n > 2.