Sums of three squares of fractions of two variables (Q1777263)

The Pythagoras number of the field \(\mathbb{R}(X,Y)\) is known to be 4 (i.e., every sum of squares is a sum of at most 4 squares, and there are sums of squares that are not sums of 3 squares). Every positive semi-definite polynomial in \(\mathbb{R}[X,Y]\) is a sum of squares in the quotient field, hence is a sum of at most four squares. The authors exhibit several new families of positive semi-definite polynomials in \(\mathbb{R}[X,Y]\) that cannot be written as sums of fewer than 4 squares in \(\mathbb{R}(X,Y)\). Their polynomials are monic of degree 4 in the variable \(Y\), the coefficients are polynomials in the variable \(X\). The coefficients are used to define certain elliptic curves over the field \(\mathbb{R}(X)\). The analysis of these curves yields criteria to decide whether or not a polynomial is a sum of 3 squares.