Capacity theory and algebraic integers
Pooh, Charles Thierry
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It is well known that there are infinitely many complete conjugate sets of algebraic integers on the unit circle and finitely many inside the unit circle. This result was generalized to arbitrary compact subsets of C by Fekete and Szeg¨o. In this work, we establish explicit effective Fekete and Fekete-Szeg¨o theorems. Specifically, let E be a compact subset of C stable under complex conjugation and such that the unbounded component of C\E is simply connected. We show that if E has logarithmic capacity .8(E) < 1 then there is a monic polynomial P (w) .Z[w], whose degree is bounded explicitly in terms of .8(E), such that the region P (w)< 1 contains an o-neighborhood of E. As a consequence, we obtain explicit bounds for the number of complete conjugate sets of algebraic integers in E, as well as explicit bounds for their heights. If .8(E)=1 and E belongs to a special class of sets such as convex sets or circular arc polygons then for each o> 0, we construct a level curve P (w)= c contained in the o-neighborhood of E. The integer c, the degree and the coefficients of the monic polynomial P (w) .Z[w] are bounded explicitly in terms of o, .8(E) and infw.E w. This shows the existence of algebraic points near E with computable “low” degree and computable heights. We prove similar results for compact sets whose outer boundaries consist of a finite number of connected components. We have also obtained results concerning the numerical approximation of logarithmic capacities, reducing the estimation of the logarithmic capacity of an arbitrary region to that of a polygonal region. We give an algorithm to approximate the logarithmic capacity with explicit pointwise estimates on the error, provided bounds on the norms of some operators are known.