7 research outputs found
The monic integer transfinite diameter
We study the problem of finding nonconstant monic integer polynomials,
normalized by their degree, with small supremum on an interval I. The monic
integer transfinite diameter t_M(I) is defined as the infimum of all such
supremums. We show that if I has length 1 then t_M(I) = 1/2.
We make three general conjectures relating to the value of t_M(I) for
intervals I of length less that 4. We also conjecture a value for t_M([0, b])
where 0 < b < 1. We give some partial results, as well as computational
evidence, to support these conjectures.
We define two functions that measure properties of the lengths of intervals I
with t_M(I) on either side of t. Upper and lower bounds are given for these
functions.
We also consider the problem of determining t_M(I) when I is a Farey
interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning
this value is true for an infinite family of Farey intervals.Comment: 32 pages, 5 figure
Intersection of algebraic plane curves: some results on the (monic) integer transfinite diameter
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 BĂ©zoutâs
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
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical
Constants (2003) and Mathematical Constants II (2019), both published by
Cambridge University Press. Comments are always welcome.Comment: 162 page
Intersection of algebraic plane curves : some results on the (monic) integer transfinite diameter
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 BeÌzoutâs 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.EThOS - Electronic Theses Online ServiceGBUnited Kingdo