Products of positive definite quadratic forms

In 1927, E. Artin solved Hilbert’s 17th problem and showed that real positive definite forms are exactly the quotients of sums of squares of forms, but did not provide any bound on the number of squares appearing in such sums.  In 1967, A. Pfister showed for every natural number n that all positive definite forms in R(X_1, …, X_n) are sums of 2^n squares.  When restricting to quadratic forms, this bound is easily seen not to be optimal.  In this talk, based on joint work with K. J. Becher, a sharper bound is presented for the products of positive definite real quadratic forms in n variables, for certain n.  

More precisely, we use methods from classical quadratic form theory to show, for any natural k, that the product of two positive definite quadratic forms in R[X_1, …, X_n], where n = 2^{k+1} + 1, is the sum of (3n-1)/2 squares of rational functions in R(X_1, …, X_n).

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062