On the spatial distribution of solutions of decomposable form equations (Q2781213)

Let \(F({\mathbf x})=F(x_1,\ldots,x_n)\in{\mathbb Z}[x_1,\ldots,x_n]\) be a decomposable form, that is a homogeneous polynomial which factorises over \({\mathbb C}\) as a product of linear forms. We assume that \(F\) has no multiple factors, and that the set of linear factors of \(F\) has rank \(n\) over \({\mathbb C}\). Then \(F\) can be factored over \({\mathbb Q}\) as \(q\prod_{i=1}^t N_{M_i/{\mathbb Q}}(\varphi_i ({\mathbf x}))\), where \(q\in{\mathbb Q}^*\) and where for \(i=1,\ldots,t\), \(\varphi_i\) is a linear form with algebraic coefficients generating the number field \(M_i\) and \(N_{M_i/{\mathbb Q}}\) denotes the norm of \(M_i/{\mathbb Q}\). NEWLINENEWLINENEWLINEThe form \(F\) is said to be of CM-type if each \(M_i\) is either a totally real field or a totally imaginary extension of a totally real field and if moreover, none of the fields \(M_i\) has a subfield of unit rank \(1\). This condition on the subfields can be omitted if \(\deg F =n\). NEWLINENEWLINENEWLINEIn what follows, \(F\) is assumed to be of CM-type. The authors consider decomposable form equations (*) \(F({\mathbf x})=a\) in \({\mathbf x}\in{\mathbb Z}^n\). They are interested in the case that (*) has infinitely many solutions. In that case, the set of solutions can be divided into a finite number of families, which may be viewed as cosets of certain finitely generated multiplicative groups. Let \(r\) denote the maximum of the ranks of these groups. For \(a\in{\mathbb Z}\) with \(a\not= 0\), denote by \(F(a,T)\) the set of \({\mathbf x}\in{\mathbb Z}^n\) such that \(F({\mathbf x})=a\) and \(|{\mathbf x}|<T\), where \(|{\mathbf x}|=\max (|x_1|,\dots,|x_n|)\). NEWLINENEWLINENEWLINEIn a previous paper, \textit{G. Everest} and \textit{K. Győry} [Acta Arith. 79, 173-191 (1997; Zbl 0883.11015))] showed that for the cardinality \(P(T)\) of \(F(a,T)\) one has an asymptotic formula of the shape NEWLINE\[NEWLINEP(T)=\rho_1(\log T)^r+\rho_2(\log T)^{r-1}+o((\log T)^{r-1})\quad \text{as}\quad T\to\infty,NEWLINE\]NEWLINE where \(\rho_1>0\) and \(\rho_2\) are constants depending only on \(a\). NEWLINENEWLINENEWLINEIn the paper under review, the authors consider the distribution of sets NEWLINE\[NEWLINEF_V(T):=\{ {\mathbf x}\in F(a,T): |{\mathbf x}|^{-1}\cdot {\mathbf x}\in V\},NEWLINE\]NEWLINE where \(V\) is a subset of the unit cube \(S:=\{ {\mathbf y}\in {\mathbb R}^n:|{\mathbf y}|=1\}\). Roughly speaking, \(F_V(T)\) consists of the central projections of the points in \(F(a,T)\) onto \(S\). Denote by \(P_V(T)\) the cardinality of \(F_V(T)\). For \(V\subset S\), denote by \(V(\varepsilon)\) the set of points in \(S\) that have distance \(<\varepsilon\) to \(V\). NEWLINENEWLINENEWLINEThe authors prove the following asymptotic estimates: NEWLINENEWLINENEWLINE1) There is a finite subset \(V\) of \(S\) such that for any \(\varepsilon >0\), NEWLINE\[NEWLINEP_{V(\varepsilon)}(T)=\rho_1(\log T)^r +O((\log T)^{r-1})\quad \text{as}\quad T\to\infty;NEWLINE\]NEWLINE 2) If \(W\) denotes the union of the projections to \(S\) of the straight lines connecting the points in \(V\), then for any \(\varepsilon >0\), NEWLINE\[NEWLINEP_{W(\varepsilon)}(T)=\rho_1(\log T)^r +\rho_2(\log T)^{r-1}+ o((\log T)^{r-1}) \quad \text{as}\quad T\to\infty;NEWLINE\]NEWLINE 3) Denote by \(V(\varepsilon)'\) the complement of \(V(\varepsilon)\) in \(S\). Then NEWLINE\[NEWLINEP_{V(\varepsilon)'}(T)=f(\varepsilon)(\log T)^{r-1}+ o((\log T)^{r-1}) \quad \text{as}\quad T\to\infty,NEWLINE\]NEWLINE where \(f(\varepsilon)=0\) for sufficiently large \(\varepsilon\) and \(\lim_{\varepsilon\downarrow 0} f(\varepsilon)=\infty\). NEWLINENEWLINENEWLINEThese results indicate that the central projections onto \(S\) of the solutions of (*) cluster around a finite number of lines, and that from the projections around these lines most are concentrated around a finite number of points. To illustrate their results, the authors have performed some numerical experiments for certain decomposable form equations.