## Intersection of algebraic plane curves: some results on the (monic) integer transfinite diameter

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##### Date

2008##### Author

Hilmar, Jan

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##### Abstract

Part I discusses the problem of determining the set of intersection points, with
corresponding multiplicities, of two algebraic plane curves. We derive an algorithm
based on the Euclidean Algorithm for polynomials and show how to use it to find the
intersection points of two given curves. We also show that an easy proof of Bézout’s
Theorem follows. We then discuss how, for curves with rational coefficients, this
algorithm can bemodified to find the intersection points with coordinates expressed
in terms of algebraic extensions of the rational numbers.
Part II deals with the problem of determining the (monic) integer transfinite
diameter of a given real interval. We show how this problem relates to the problem
of determining the structure of the spectrum of normalised leading coefficients of
polynomials with integer coefficients and all roots in the given interval. We then
find dense regions of this spectrum for a number of intervals and discuss algorithms
for finding discrete subsets of the spectrum for the interval [0,1]. This leads to an
improvement in the known upper bound for the integer transfinite diameter. Finally,
we discuss the connection between the infimum of the spectrum and the monic
integer transfinite diameter.