Estimates for polynomials in logarithms of some rational numbers
The Hermite-Padé approximations of the second type for an algebra generated by a generalized Nikishin system of Markov functions corresponding to an infinite branching graph are investigated. Arithmetical applications of this construction are given. Namely, lower estimates for polynomials with integer coefficients in logarithms of some rational numbers are obtained. These estimates partially refine some known results obtained earlier by the Siegel method. The author proves the following. Theorem 1. If \(r, p\), and \(q\) are natural numbers which satisfy the condition \(q > \kappa p^{N+1}\), where \(\kappa = \kappa_r(p) = 6^{rN}2^{r+N}\) and \(N = N_r(p) = (p+1)^r-1\), then, for any nonzero polynomial \(P\) of \(p\) variables which have integer rational coefficients and whose degree does not exceed \(r\), i.e., \(P(z_1, \ldots, z_p)\in\mathbb Z[z_1, \ldots, zp]\), \(P\not\equiv 0\), \(\deg P\leq r\), the following inequality holds: \[ | P(\theta_1,\ldots, \theta_p)| > \frac{c}{q^N}\frac 1{H^\mu}, \] where \(H = H(P)\) is the height of polynomial \(P\), exponent \(\mu\) is defined by the formula \[ \mu=\mu_r(p;q)=\frac{N\log q+\log\kappa}{\log q-\log(\kappa p^{N+1})}, \] and \(c\) is an effective positive constant depending only on the numbers \(r\) and \(p\).
Hermite-Padé approximations of the second type
generalized Nikishin system of Markov functions
scientific article; zbMATH DE number 5358124