Arturas Dubickas

Polynomials Irreducible by Eisenstein's Criterion

Abstract.We compute the probability for a monic univariate integer polynomial to be irreducible by Eisensteins Criterion. In particular, it follows that less than 1% of the polynomials with at least seven non-zero coefficients are irreducible by Eisenstein.

VARIATIONS ON THE THEME OF HILBERT'S THEOREM 90

Given an algebraic number field K, we find two separate necessary and sufficient conditions on a given algebraic number for it to be expressible as a quotient, or as a difference, of two algebraic numbers which are conjugate over K.

The conjugate dimension of algebraic numbers

We find sharp upper and lower bounds for the degree of an algebraic number in terms of the Q-dimension of the space spanned by its conjugates. For all but seven non-negative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2 n n!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of GL n(Q); this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4. We extend our results in two directions. We consider the problem when Q is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension Q(ω� ) of Q. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number.

On the lines passing through two conjugates of a Salem number

We show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number α of degree d 3i s at most[d 2 /2 ]− d + 2, its conjugates being in general position if this number is attained. If, for instance, d 4 is even, then the conjugates of α ∈ Q of degree d are in general position if and only if α has 2 real conjugates, d − 2 complex conjugates, no three distinct conjugates of α lie on a line and any two lines that pass through two distinct conjugates of α are non-parallel, except for d/2 − 1 lines parallel to the imaginary axis. Our main result asserts that the conjugates of any Salem number are in general position. We also ask two natural questions about conjugates of Pisot numbers which lead to the equation α1 + α2 = α3 + α4 in distinct conjugates of a Pisot number. The Pisot number α1 = (1 + 3 + 2 √ 5)/2 shows that this equation has such a solution.

The Remak height for units

We investigate the values of the Remak height, which is a weighted product of the conjugates of an algebraic number. We prove that the ratio of logarithms of the Remak height and of the Mahler measure for units αof degree d is everywhere dense in the maximal interval [d/2(d-1),1] allowed for this ratio. To do this, a “large” set of totally positive Pisot units is constructed. We also give a lower bound on the Remak height for non-cyclotomic algebraic numbers in terms of their degrees. In passing, we prove some results about some algebraic numbers which are a product of two conjugates of a reciprocal algebraic number.

No two non-real conjugates of a Pisot number have the same imaginary part

We show that the number α = (1 + √ 3 + 2 √ 5)/2 with minimal polynomial x4 − 2x3 + x − 1 is the only Pisot number whose four distinct conjugates α1, α2, α3, α4 satisfy the additive relation α1+α2 = α3+α4. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations α1 = α2 + α3 + α4 or α1 + α2 + α3 + α4 = 0 cannot be solved in conjugates of a Pisot number α. We also show that the roots of the Siegel’s polynomial x3−x−1 are the only solutions to the three term equation α1+α2+α3 = 0 in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation α1 = α2 + α3.

Lower bounds for Z-numbers

Let p/q be a rational noninteger number with p > q ≥ 2. A real number λ ≥ 0 is a Z p/q -number if {λ(p/q) n } < 1/q for every nonnegative integer n, where {x} denotes the fractional part of x. We develop several algorithms to search for Z p/q -numbers, and use them to determine lower bounds on such numbers for several p and q. It is shown, for instance, that if there is a Z 3/2 -number, then it is greater than 2 57 . We also explore some connections between these problems and some questions regarding iterated maps on integers.

On the number of integer polynomials with multiplicatively dependent roots

We give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for various cases, although in general there is a logarithmic gap between lower and upper bounds.