##
**The number of solutions of polynomial-exponential equations.**
*(English)*
Zbl 0949.11020

Let \(P_l\) \((l=1,\ldots ,k)\) be non-zero polynomials in \(K[X_1,\ldots ,X_n]\) where \(K\) is an algebraic number field. Let \({\mathbf a}_l=(a_{1l},\ldots ,a_{nl})\) \((l=1,\ldots ,n)\) be vectors with coordinates in \(K^*\), and write \({\mathbf a_l}^{{\mathbf x}}=a_{l1}^{x_{l1}}\cdots a_{ln}^{x_{ln}}\) for \({\mathbf x}=(x_1,\ldots ,x_n)\in{\mathbb Z}^n\). The authors deal with equations of the shape
\[
\sum_{l=1}^k P_l({\mathbf x}){\mathbf a}_l^{{\mathbf x}}=0\quad \text{in \({\mathbf x}\in{\mathbb Z}^n\).} \tag{1}
\]
In the special case \(n=1\) this is equivalent to finding the zeros of a linear recurrence equation. In the 1980’s, M. Laurent proved a general result on (1), a special case of which is that (1) has only finitely many non-degenerate solutions [Astérisque 147-148, 121-139 (1987; Zbl 0621.10014)]. Laurent’s proof goes back to the Subspace Theorem. Then the authors derived an explicit upper bound for the number of solutions of (1) depending on \(n\), the degree of \(K\), the total degrees of the polynomials \(P_l\), and the number of prime ideals occurring in the prime ideal decompositions of the \(a_{lj}\) [Math. Ann. 296, 339-361 (1993; Zbl 0805.11029)]. In the present paper, the authors derive a much better bound, which has no dependence anymore on the prime ideal factorizations of the \(a_{lj}\) and which has a much better dependence on the other parameters mentioned above.

For any partition \({\mathcal P}=\{ \lambda_1,\ldots ,\lambda_r\}\) of \(\{ 1,\ldots ,k\}\) into pairwise disjoint non-empty sets, let \(G({\mathcal P})\) be the subgroup of \({\mathbb Z}^n\) generated by all vectors \({\mathbf a}_i-{\mathbf a}_j\) \((i,j\in\lambda_t, t=1,\ldots r)\). A solution \({\mathbf x}\) of (1) is called degenerate if there is a partition \({\mathcal P}=\{ \lambda_1,\ldots ,\lambda_r\}\) such that \(\sum_{l\in \lambda_t} P_l({\mathbf x}){\mathbf a}_l^{{\mathbf x}}=0\) for \(t=1,\ldots ,r\) and \(G({\mathcal P})\not=({\mathbf 0})\), and otherwise non-degenerate. The new result of the authors is as follows: let \(A=\sum_{l=1}^k {n+\delta_l\choose n}\) and \(B=\max (n,A)\), where \(\delta_l\) is the total degree of \(P_l\), and let \(d=[K:{\mathbb Q}]\); then equation (1) has at most \(2^{36B^3}d^{6B^2}\) non-degenerate solutions. More generally, the authors prove a similar result for systems of equations (1) instead of just a single equation.

Part of the improvement of the upper bound was an automatic consequence of new versions of the quantitative Subspace Theorem due to Schlickewei (which made it possible to get rid of the dependence on the prime ideal factorization of the \(a_{lj}\)’s) and from an improvement of Roth’s lemma by the reviewer based on the proof of Faltings’ Product Theorem. Combined with the arguments with which the authors deduced their previous bound this would have given an upper bound for the number of non-degenerate solutions of (1) doubly exponential in \(B\). But in the present paper the authors develop a new argument which yields a reduction to a bound exponential in a power of \(B\).

Basically, one has to rewrite (1) as \(\sum_{m=1}^N a_my_m({\mathbf x})=0\), where \(a_m\in K^*\) is a constant and where each \(y_m({\mathbf x})\) is a term of the shape \(M({\mathbf x}){\mathbf a}^{{\mathbf x}}\), with \(M({\mathbf x})\) a monomial in \(x_1,\ldots ,x_n\) and with \({\mathbf a}\in\{ {\mathbf a}_1,\ldots ,{\mathbf a}_k\}\). A major problem is to handle the coefficients \(a_m\). In their previous paper, the authors eliminated the \(a_m\) by taking \(N\) solutions \({\mathbf x}_1,\ldots {\mathbf x}_N\) and expanding the determinant of \((y_m({\mathbf x}_p))_{m,p=1,\ldots ,N}\). This led to an equation in \(nN\) variables with \(N!\) terms and with coefficients \(\pm 1\).

In the present paper, instead of using this uneconomical determinant argument, the authors use that if one applies a transformation of the shape \({\mathbf x}\mapsto {\mathbf u}+A{\mathbf x}\) to the set of solutions of (1), where \({\mathbf u}\in{\mathbb Z}^n\) and \(A\in GL (n,{\mathbb Z})\), then one gets another equation of type (1) with the same number of non-degenerate solutions. Then using ingeneous techniques from geometry of numbers they show that one can choose such a transformation in such a way that after an application of this, the sizes of the coefficients \(a_m\) are relatively small compared with the sizes of the solutions.

Some work lying ahead is to deduce an explicit upper bound for the number of non-degenerate solutions of (1) which is independent of \(d=[K:{\mathbb Q}]\), i.e., which depends only on the “unavoidable” parameters \(k,n\) and the \(\delta_l\). Thanks to a new version of the quantitative Subspace Theorem due to Schlickewei and the reviewer, this could be established in certain special cases, but then there was a trade-off in the sense that eliminating \(d\) led to a much worse dependence on the other parameters. First, in the case that \(n\geq 1\) and all polynomials \(P_l\) are constants, Schlickewei, Schmidt and the reviewer deduced an explicit upper bound for the number of non-degenerate solutions of (1) depending exponentially on \(n\) and doubly exponentially on \(k\). Then W. M. Schmidt [Acta Math. 182, No. 2, 243-282 (1999)], using apart from this new version of the quantitative Subspace Theorem various other new techniques developed by himself, gave in the case \(n=1\) an explicit upper bound for the number of solutions of (1) which is triply exponential in \(\sum_{l=1}^k \delta_l\).

For any partition \({\mathcal P}=\{ \lambda_1,\ldots ,\lambda_r\}\) of \(\{ 1,\ldots ,k\}\) into pairwise disjoint non-empty sets, let \(G({\mathcal P})\) be the subgroup of \({\mathbb Z}^n\) generated by all vectors \({\mathbf a}_i-{\mathbf a}_j\) \((i,j\in\lambda_t, t=1,\ldots r)\). A solution \({\mathbf x}\) of (1) is called degenerate if there is a partition \({\mathcal P}=\{ \lambda_1,\ldots ,\lambda_r\}\) such that \(\sum_{l\in \lambda_t} P_l({\mathbf x}){\mathbf a}_l^{{\mathbf x}}=0\) for \(t=1,\ldots ,r\) and \(G({\mathcal P})\not=({\mathbf 0})\), and otherwise non-degenerate. The new result of the authors is as follows: let \(A=\sum_{l=1}^k {n+\delta_l\choose n}\) and \(B=\max (n,A)\), where \(\delta_l\) is the total degree of \(P_l\), and let \(d=[K:{\mathbb Q}]\); then equation (1) has at most \(2^{36B^3}d^{6B^2}\) non-degenerate solutions. More generally, the authors prove a similar result for systems of equations (1) instead of just a single equation.

Part of the improvement of the upper bound was an automatic consequence of new versions of the quantitative Subspace Theorem due to Schlickewei (which made it possible to get rid of the dependence on the prime ideal factorization of the \(a_{lj}\)’s) and from an improvement of Roth’s lemma by the reviewer based on the proof of Faltings’ Product Theorem. Combined with the arguments with which the authors deduced their previous bound this would have given an upper bound for the number of non-degenerate solutions of (1) doubly exponential in \(B\). But in the present paper the authors develop a new argument which yields a reduction to a bound exponential in a power of \(B\).

Basically, one has to rewrite (1) as \(\sum_{m=1}^N a_my_m({\mathbf x})=0\), where \(a_m\in K^*\) is a constant and where each \(y_m({\mathbf x})\) is a term of the shape \(M({\mathbf x}){\mathbf a}^{{\mathbf x}}\), with \(M({\mathbf x})\) a monomial in \(x_1,\ldots ,x_n\) and with \({\mathbf a}\in\{ {\mathbf a}_1,\ldots ,{\mathbf a}_k\}\). A major problem is to handle the coefficients \(a_m\). In their previous paper, the authors eliminated the \(a_m\) by taking \(N\) solutions \({\mathbf x}_1,\ldots {\mathbf x}_N\) and expanding the determinant of \((y_m({\mathbf x}_p))_{m,p=1,\ldots ,N}\). This led to an equation in \(nN\) variables with \(N!\) terms and with coefficients \(\pm 1\).

In the present paper, instead of using this uneconomical determinant argument, the authors use that if one applies a transformation of the shape \({\mathbf x}\mapsto {\mathbf u}+A{\mathbf x}\) to the set of solutions of (1), where \({\mathbf u}\in{\mathbb Z}^n\) and \(A\in GL (n,{\mathbb Z})\), then one gets another equation of type (1) with the same number of non-degenerate solutions. Then using ingeneous techniques from geometry of numbers they show that one can choose such a transformation in such a way that after an application of this, the sizes of the coefficients \(a_m\) are relatively small compared with the sizes of the solutions.

Some work lying ahead is to deduce an explicit upper bound for the number of non-degenerate solutions of (1) which is independent of \(d=[K:{\mathbb Q}]\), i.e., which depends only on the “unavoidable” parameters \(k,n\) and the \(\delta_l\). Thanks to a new version of the quantitative Subspace Theorem due to Schlickewei and the reviewer, this could be established in certain special cases, but then there was a trade-off in the sense that eliminating \(d\) led to a much worse dependence on the other parameters. First, in the case that \(n\geq 1\) and all polynomials \(P_l\) are constants, Schlickewei, Schmidt and the reviewer deduced an explicit upper bound for the number of non-degenerate solutions of (1) depending exponentially on \(n\) and doubly exponentially on \(k\). Then W. M. Schmidt [Acta Math. 182, No. 2, 243-282 (1999)], using apart from this new version of the quantitative Subspace Theorem various other new techniques developed by himself, gave in the case \(n=1\) an explicit upper bound for the number of solutions of (1) which is triply exponential in \(\sum_{l=1}^k \delta_l\).

Reviewer: Jan-Hendrik Evertse (Leiden)