Naslov
Polynomial root separation and applications

Autori
Pejković, Tomislav

Vrsta, podvrsta i kategorija rada
Ocjenski radovi, doktorska disertacija

Fakultet
Prirodoslovno-matematički fakultet, Matematički odsjek

Mjesto
Zagreb ; Strasbourg

Datum
21.01

Godina
2012

Stranica
81

Mentor
Bugeaud Yann ; Dujella, Andrej

Ključne riječi
integer polynomials ; root separation ; p-adic numbers ; transcendental numbers ; Mahler's classification ; Koksma's classification

Sažetak
In this thesis we study bounds on the distances of roots of integer polynomials and applications of such results. Denote by sep(P) the minimal distance of roots of the separable integer polynomial P(X) and by H(P) maximum of the absolute values of its coeficients. In the rst chapter which looks at polynomial roots in the set of complex numbers, we first summarize results on quadratic and cubic polynomials. The bulk of this chapter is dedicated to quartic polynomials and especially reducible monic integer polynomials of fourth degree. We show that for such polynomials sep(P) >> H(P)^(-2) but also construct families (P_k(X)) of such polynomials that have sep(P) ~ H(P)^(-2). The case when coeficients of Pk(X) are polynomials in k is studied more thoroughly. In the second chapter different lemmas on roots of polynomials in the p-adic setting are proved. These lemmas are mostly analogues of the results in the real and complex case and are used later in the thesis. In the third chapter explicit families of polynomials of general degree n are given which bound the exponent above H(P) from the other side than sep_p(P) >> H(P)^(-n+1). Results are proved using Newton polygons. Then the case of quadratic and reducible cubic polynomials in the p-adic setting is completely solved which shows that the bound above is really attained in those classes of polynomials. For irreducible cubic polynomials a bound with a new, better exponent is exhibited. The rest of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w n and the related classifications of transcendental numbers in C_p. In the fourth chapter the main result is a construction of numbers such that the two functions w_n and w*_n differ on them for every n. We can even require w_n - w*_n to be a chosen number in some small interval. The proof is quite involved and follows R. C. Baker's proof in the real case. In the fifth chapter the interval of possible values for w_n - w*_n is expanded using an effective estimate for the distance of algebraic numbers and a family of polynomials with very close roots. The main proof in this chapter follows the one in the previous chapter, but is a little easier since we restrict ourselves to one or finitely many n. Special attention is given to cases n = 1 and n = 2. In the last chapter inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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