An inequality for values of Koksma's functions of two algebraically dependent \(p\)-adic numbers
For \(p\) be a prime number, let \(\mathbb{Q}_p\) denote the completion of the field of rational numbers \(\mathbb{Q}\) with respect to \(p\)-adic absolute value \(| \cdot |_p\). Denote by \(w_n\) Mahler's function on \(\mathbb{Q}_p\) and denote by \(w_n^*\) Koksma's function on \(\mathbb{Q}_p\). The author shows that if \(\xi\), \(\eta\in \mathbb{Q}_p\) are two algebraically dependent transcendental numbers and \(P(\xi, \eta)=0\), where \(P(x, y)\in \mathbb{Z}[x,y]\) is a non-zero irreducible polynomial over \(\mathbb{Q}\), then \(w_n^*(\xi)+1\leq M(w^*_{nN}(\eta)+1)\) and \(w_n^*(\eta)+1\leq N(w^*_{nM}(\xi)+1)\) for every positive integer \(n\). Moreover it is shown that these inequalities become equalities in a special case.
scientific article; zbMATH DE number 6806128