On the number of solutions of decomposable polynomial equations (Q2773329)

By a decomposable polynomial of degree \(n\) in \(m\) variables with coefficients in a field \(K\) we mean a polynomial \(F({\mathbf X})=F(X_1,\ldots,X_m)\in K[X_1,\ldots,X_m]\) that can be expressed as a product \(\prod_{i=1}^n (\alpha_{i1}X_1+\cdots +\alpha_{im}X_m+\lambda_i)\) with the \(\alpha_{ij}\), \(\lambda_i\) lying in the algebraic closure of \(K\). If all \(\lambda_i\) are \(0\), then \(F\) is called a decomposable form. Let \(S=\{ p_1,\ldots,p_s\}\) be a finite set of \(s\geq 0\), primes and let \({\mathbb{Z}}_S={\mathbb{Z}}[(p_1\cdots p_s)^{-1}]\) denote the ring of \(S\)-integers. The authors consider equations NEWLINE\[NEWLINEF({\mathbf x})=b\quad\text{in }{\mathbf x}\in{\mathbb{Z}}_S^m \tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEF({\mathbf x})\in b{\mathbb{Z}}_S^*\quad\text{in }{\mathbf x}\in{\mathbb{Z}}_S^m,\tag{2}NEWLINE\]NEWLINE where \({\mathbb{Z}}_S^*\) denotes the unit group of \({\mathbb{Z}}_S\). In 1989, \textit{I. Gaál, K. Győry} and the reviewer [Arch. Math. 52, 337-353 (1989; Zbl 0655.10017)] gave sufficient conditions for equations (1), (2) to have only finitely many solutions. In the present paper, the authors give explicit upper bounds for the number of solutions of equations (1), (2). To give a flavour, we mention two consequences of their general results. Let \(b\in{\mathbb{Z}}_S\), \(b\not= 0\). Denote by \(\omega_S(b)\) the number of primes \(p\) outside \(S\) that divide \(b\) in \({\mathbb{Z}}_S\). Let \(M\) be a number field of degree \(n\) and \(\xi\mapsto \xi^{(i)}\) \((i=1,\ldots,n)\) the isomorphisms of \(M\) into \({\mathbb{C}}\). Let \(L({\mathbf X})=\sum_{k=1}^m\alpha_kX_k+\lambda\) with \(\alpha_1,\ldots,\alpha_m,\lambda\in M\). We obtain \(L^{(j)}({\mathbf X})\) by applying \(\xi\mapsto \xi^{(j)}\) to the coefficients of \(L\). NEWLINENEWLINENEWLINE(1) Suppose \(M\) has no proper subfield. Suppose that \(1,\alpha_1,\ldots,\alpha_m,\lambda\) are linearly independent over \({\mathbb{Q}}\). Let \(a_0\in {\mathbb{Z}}_S\), \(a_0\not= 0\) such that the polynomial \(a_0D_{M/{\mathbb{Q}}}(L({\mathbf X}))= a_0\prod_{1\leq i<j\leq n}(L^{(i)}({\mathbf X})-L^{(j)}({\mathbf X}))^2\) has its coefficients in \({\mathbb{Z}}_S\). Then the equation \(a_0D_{M/{\mathbb{Q}}}(L({\mathbf x}))\in b{\mathbb{Z}}_S^*\) has at most \(\big(2^{17}n(n-1)\big)^{\delta (m)(s+\omega_S(b)+1)}\) solutions in \({\mathbf x}\in{\mathbb{Z}}_S^m\), where \(\delta (m)={2\over 3}(m+1)(m+2)(2m+3)-4\). NEWLINENEWLINENEWLINE(2) Suppose that \(\alpha_1,\ldots,\alpha_m,\lambda\) are linearly independent over \({\mathbb{Q}}\). Let \(a_0\in {\mathbb{Z}}_S\), \(a_0\not= 0\) such that the polynomial \(a_0N_{M/{\mathbb{Q}}}(L({\mathbf X}))=a_0\prod_{j=1}^n L^{(j)}({\mathbf X})\) has its coefficients in \({\mathbb{Z}}_S\). Then the equation \(a_0N_{M/{\mathbb{Q}}}(L({\mathbf x}))\in b{\mathbb{Z}}_S^*\) has at most \(\big(2^{17}n\big)^{\delta (m)(s+\omega_S(b)+1)}\) solutions in \({\mathbf x}\in{\mathbb{Z}}_S^m\). NEWLINENEWLINENEWLINEThe authors deduce their results from a theorem of \textit{K. Győry} and the reviewer [Acta Arith. 80, 367-394 (1997; Zbl 0886.11015)], giving an explicit upper bound for the number of families of solutions of decomposable form equations (i.e. with all \(\lambda_i=0\)).