Further variations on the six exponentials theorem. (English) Zbl 1116.11054

Waldschmidt, Michel

Hardy-Ramanujan J. 28, 1-9 (2005).

Let \(\mathbb{Q}\) be the field of rational numbers, \(\overline{\mathbb{Q}}\) the field of algebraic numbers and \(\overline{\mathcal L}\) the \(\overline{\mathbb{Q}}\)-vector subspace of \(\mathbb{C}\) spanned by \(1\) and the logarithms of algebraic numbers. The main result of the paper is the following: “Let \(M\) be a \(2\times 3\) matrix with entries in \(\overline{\mathcal L}\) \[ M= \begin{pmatrix} \Lambda_{11} &\Lambda_{12} &\Lambda_{13}\\ \Lambda_{21} &\Lambda_{22} &\Lambda_{23}\end{pmatrix}. \] Assume that the five rows of the matrix \[ \begin{pmatrix} M\\ I_3\end{pmatrix}= \begin{pmatrix} \Lambda_{11} &\Lambda_{12} &\Lambda_{13}\\ \Lambda_{21} &\Lambda_{22} &\Lambda_{23}\\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix} \] are linearly independent over \(\overline{\mathbb{Q}}\) and that the five columns of the matrix \[ (I_2, M)= \begin{pmatrix} 1 & 0 & \Lambda_{11} & \Lambda_{12} & \Lambda_{13}\\ 0 & 1 & \Lambda_{21} & \Lambda_{22} & \Lambda_{23}\end{pmatrix} \] are linearly independent over \(\overline{\mathbb{Q}}\). Then one at least of the three numbers \[ \Delta_1= \left|\begin{matrix} \Lambda_{12} &\Lambda_{13}\\ \Lambda_{22} & \Lambda_{23}\end{matrix}\right|,\quad \Delta_2= \left|\begin{matrix} \Lambda_{13} & \Lambda_{11}\\ \Lambda_{23} & \Lambda_{21}\end{matrix}\right|,\quad \Delta_3= \left|\begin{matrix} \Lambda_{11} & \Lambda_{12}\\ \Lambda_{21} & \Lambda_{22}\end{matrix}\right| \] is not in \(\overline{\mathcal L}\).
The proof depends on the following generalization of Proposition 6.1 of the author paper “Transcendance et exponentielles en plusieurs variables”, Invent. Math. 63, 97–127 (1981; Zbl 0454.10020):
Let \(X\) and \(Y\) be two \(\overline{\mathbb{Q}}\)-vector subpaces of \(\mathbb{C}^n\). Assume \(X\) has dimension \(d\) with \(d> n\). Assume further \(\mu(Y,\mathbb{C}^n)> {d\over d-n}\). Then the set \(X\cdot Y\) is not contained in \(\overline{\mathcal L}\). Here this result is applied to the following situation: Let \(v_1= (1,0)\), \(v_2= (0,1)\), \(v_{2+j}= (\Lambda_{1j}, \Lambda_{2j})\), \(j= 1,2,3\). For \(v= (x,y)\in\mathbb{C}^2\) set \(v'= (-y, x)\). Let \(X\) be the \(\overline{\mathbb{Q}}\)-vector subspace of \(\mathbb{C}^2\) spanned by \(v_1,\dots,v_5\) and \(Y\) that one spanned by \(v_1',\dots, v_5'\). The author can show that for these spaces \(X\) and \(Y\) the assumptions in his generalized Proposition 1 are fulfilled. Consequently the set \(X\cdot Y\) is not in \(\overline{\mathcal L}\) and from this one concludes easily that at least one of \(\Delta_1\), \(\Delta_2\), \(\Delta_3\) is not in \(\overline{\mathcal L}\). Several corollaries are given. The author expects that a similar result holds for a \(2\times 2\) matrix \(M\).

Reviewer: Rolf Wallisser (Freiburg i. Br.)

MSC:

11J81	Transcendence (general theory)
11J85	Algebraic independence; Gel’fond’s method
11J86	Linear forms in logarithms; Baker’s method

Keywords:

six exponentials theorem; rank of matrices with coefficients being linear forms in logarithm

Citations:

Zbl 0454.10020