A paucity estimate related to Newton sums of odd degree (Q2902657)

Consider the system of equations NEWLINE\[NEWLINE\sum_{i=0}^kx_i^{2j-1}=\sum_{i=0}^ky_i^{2j-1}\;\;\;(1\leq j\leq k)NEWLINE\]NEWLINE in which one has positive integer variables \(x_0,\ldots,x_k, y_0,\ldots,y_k\leq B\). This paper gives bounds for the number \(U_k(B)\) of non-trivial solutions (in which the \(x_i\)'s are not a permutation of the \(y_i\)'s.) Any such solution leads to a non-trivial solution to the ``multigrade'' system NEWLINE\[NEWLINE\sum_{i=0}^mX_i^j=\sum_{i=0}^mY_i^j\;\;\;(1\leq j\leq k')NEWLINE\]NEWLINE with \(k'=2k\) and \(m=k'+1\). \textit{R. C. Vaughan} and \textit{T. D. Wooley} [Acta Arith. 79, No. 3, 193--204 (1997; Zbl 0887.11042)] have shown that there are \(O_k(B^{\sqrt{4k'+5}})\) non-trivial solutions for such a system when \(m=k'\), but no such ``paucity'' estimate is available when \(m=k'+1\).NEWLINENEWLINEThe present paper shows that \(U_k(B)\ll_{k,\varepsilon}B^{\lambda(k)+\varepsilon}\) for any \(\varepsilon>0\), where the exponent \(\lambda(k)\) is strictly less than \(k+1\). For comparison we note that the number of trivial solutions is asymptotically \((k+1)!B^{k+1}\). It is shown further that one may take \(\lambda(3)=\tfrac{34}{9}\), and \(\lambda(k)=\tfrac{6}{7}k+O(1)\) in general.NEWLINENEWLINEThe proof begins by showing that all solutions satisfy further conditions, such as NEWLINE\[NEWLINE\prod_{0\leq i<j\leq k}(x_i+x_j)=\prod_{0\leq i<j\leq k}(y_i+y_j).NEWLINE\]NEWLINE Many such product formulae are used. The factorization information obtained from these, presumably corresponding to intermediate torsors for the original variety, is then combined in an ingenious ad hoc way to give the bounds quoted above.