Local positivity and effective Diophantine approximation (Q2110262)

The author presents a new approach to prove effective results in Diophantine approximation. Among others he proves that if \(\alpha_1,\alpha_2\) are algebraic numbers in a number field of degree \(d\) and these numbers and all of their conjugates are nonsingular points of an irreducible curve of degree \(m\) defined over \(\mathbb{C}\), then for all \(\delta\in\mathbb{Q}\) with \(\delta > \max(m,d/m)\) there exists an effectively computable constant \(C_0(\alpha_1,\alpha_2,\delta,m)\) such that for all pairs of rational numbers \((p_1/q, p_2/q)\) satisfying \[ \left|\alpha_i-\frac{p_i}{q}\right|\le q^{-\delta}\text{ for } i=1,2 \] we have \(q\le C_0(\alpha_1,\alpha_2,\delta,m)\).