Squarefree density of polynomials (Q6595584)

The paper under review inspired by a talk given by \textit{M. Bhargava} [``The geometric sieve and the density of squarefree values of invariant polynomials'', Preprint, \url{arXiv:1402.0031}], concerning the square-free values of the following integral forms of degree \(d\) in \(s\ge 2\) variables \(\mathbf x=(x_1,\ldots,x_s)\), in various special cases,\N\[\N\sum_{\substack{i_1,\ldots,i_d\\\Ni_1\le \cdots\le i_d\le s}} c_{i_1\cdots i_d} x_{i_1}\cdots x_{i_d}.\N\]\NThe main result of the present paper was intended as part of an attack on the cubic case \(\mathcal C(\mathbf x)\) not of the shape \(a(b_1x_1+\cdots+b_sx_s)^3\) where \(a,b_1,\ldots,b_s\in\mathbb Q\), for which, the authors show that letting \(M(P)\) to be the number of solutions of \(\mathcal C(\mathbf x) =\mathcal C(\mathbf X)\) with \(\mathbf x, \mathbf X\in[-P,P]^s\), one has the following best possible (apart possibly from the \(\varepsilon\)) approximation,\N\[\NM(P)\ll P^{2s-2+\varepsilon}.\N\]\NNot restricted by forms, the authors consider general integral polynomials \(\mathcal P(\mathbf x) \in\mathbb Z[x_1,\ldots,x_s]\) over \(|x_j|\le P_j\) (for each \(j\le s\)), and letting\N\[\N\mathfrak S_{\mathcal P} = \prod_p \left( 1-\frac{\rho_{\mathcal P}(p^2)}{p^{2s}} \right)\N\]\Nwith \(\rho_{\mathcal P}(d) = \#\{\mathbf x\in \mathbb Z_d^s: \mathcal P(\mathbf x)\equiv 0\pmod{d}\}\), they show that for \(s\geq 2\), if \(\mathfrak S_{\mathcal P}\) is divergence, then as \(\min_jP_j\to\infty\) one has\N\[\N\sum_{\substack{\mathbf x\\\N|x_j|\le P_j}} \mu\big(|\mathcal P(\mathbf x)|\big)^2=o(P_1\ldots P_s),\N\]\Nwhere \(\mu\) is the Möbius function with taking \(\mu(0)=0\). The authors also study the special case \(\mathcal P(\mathbf x) = cx_1^k + \mathcal C^*(x_2,\ldots,x_s)\), with supposing that \(s\ge 3\), \(k=3\) or \(4\), \(c\in\mathbb Z\setminus\{0\}\) and the integral cubic form \(\mathcal C^*(x_2,\ldots,x_s)\) is not of the shape \(a(b_2x_2+\cdots+b_sx_s)^3\) where \(a,b_1,\ldots,b_s\in\mathbb Q\). Letting \(P\) is large and \(Q=P_1=P^{3/k}\), \(P_j=P\) \((2\le j\le s)\), then they show that as \(P\to\infty\) one has\N\[\N\sum_{|x_1|\le Q} \sum_{\substack{x_2,\ldots,x_s\\\N|x_j|\le P}} \mu\big(|\mathcal P(\mathbf x)|\big)^2= 2^sQP^{s-1}\mathfrak S_{\mathcal P} + E_k,\N\]\Nwith \(E_3\ll P^{s-\frac1{8s-2}+\varepsilon}\) and \(E_4\ll QP^{s-1}/((\log P)(\log\log P))\).