Continuous Pythagoras numbers for rational quadratic forms

AbstractWe 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 > 1. However, if only the product of the weight functions and the coefficient functions is required to be continuous, then n ≤ Pc(Ln(R)) < [n!e] (where e is the base of the natural logarithms) and 2 < Pc(L2(R)); we conjecture that n < Pc(Ln(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 Pc(L2(Q)) = 2, and we conjecture that Pc(Ln(Q)) = n also for n > 2