Continuous Pythagoras numbers for rational quadratic forms

We introduce the concept of the continuous Pythagoras number Pc(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 Pc(Q) = 4. Letting Ln(F) be the set of linear n-ary forms over the field F, we show that Pc(Ln(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 Pc(L1(R)) = 1 and Pc(Ln(R)) = ∞ for n \u3e 1. However, if only the product of the weight functions and the coefficient functions is required to be continuous, then n ≤ Pc(Ln(R)) \u3c [n!e] (where e is the base of the natural logarithms) and 2 \u3c Pc(L2(R)); we conjecture that n \u3c Pc(Ln(R)) also for n \u3e 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 Pc(L2(Q)) = 2, and we conjecture that Pc(Ln(Q)) = n also for n \u3e 2. © 1987