An elementary proof that the length of \(X_1^4+X_2^4+X_3^4+X_4^4\) is 4. (Q1877768)

This note contains an elementary proof that the quartic form \(X_1^4+X_2^4+X_3^4+X_4^4\) cannot be written as a sum of three squares of real quadratic forms. A proof of this result using a geometric argument was previously given by \textit{P. Yiu} [J. Pure Appl. Algebra 156, No. 2--3, 367--373 (2001; Zbl 0997.11034)]. More generally, it has been conjectured by \textit{B. Reznick} [J. Reine. Angew. Math. 377, 108--112 (1987; Zbl 0615.10028)] that for any positive integer \(k\) the form \(X_1^{2^k}+X_2^{2^k}+\dots +X_{2^k}^{2^k}\) cannot be written as a sum of fewer than \(2^k\) squares of homogeneous polynomials over \(\mathbb R\).