Paucity phenomena for polynomial products
Let P(x)∈Z[x] be a polynomial with at least two distinct complex roots. We prove that the number of solutions (x1,…,xk,y1,…,yk)∈[N]2k to the equation
∏1≤i≤kP(xi)=∏1≤j≤kP(yj)≠0
(for any k≥1 ) is asymptotically k!Nk as N→+∞. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums 1N√∑n≤Nf(P(n)) match standard complex Gaussian moments as N→+∞
, where f is the Steinhaus random multiplicative function.
London Mathematical Society