Reciprocal Polynomials With Small House

The house of an algebraic integer is the maximal modulus of its all conjugates, denoted by (?). We write this thesis, due to the Schinzel and Zassenhaus' Conjecture: There is an absolute constant c₁> 1, such that if (?) > 1, then (?)≥1 + c₁/d, where d is the degree ofα. Many theoretical lower bound had been obtained by different people since then. One important result is due to Smyth in 1971, for the nonreciprocal case. He proved that ifαis a nonreciprocal algebraic integer of degree d > 1, then (?)≥1 + logθ₀/d, whereθ₀= 1.3247... is the real root of X³-X-1.In 1985, Boyd[Bo85] gave a algorithm to search for the smallest house of algebraic integers of degree≤12, and reciprocal algebraic integers of degree≤16. In 2007, Rhin and Wu[RW] extended his computational search to d = 29.In this thesis, we use an algorithm to search for reciprocal algebraic integers with small house of degree≤26[FLW]. Our computation use a method concerned the theory of integer transfinite diameter, auxiliary function and semi-linear programming.These tools are used to give good bounds on the coefficients of the minimal polynomial of reciprocalα.In this work, we discuss also the diophantine equation and give all the integer solutions of X³Â±8 = 13Y²[Fang].