On the height of the Sylvester Resultant

The height of the mixed sparse resultant

We present an upper bound for the height of the mixed sparse resultant, defined as the logarithm of the maximum modulus of its coefficients. We obtain a similar estimate for its Mahler measure.

Endomorphisms of Bounded Height and Resultant

Let K be an algebraic number field. For a degree d rational morphism of projective n-space defined over K let R denote its minimal resultant ideal. For a fixed height function on the moduli space of dynamical systems this paper shows that all such morphisms of bounded height and resultant are contained in finitely many PGL(K) equivalence classes. This answers a question of Silverman in the affirmative.

The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials

A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials. The \emph{resultant} of two polynomials is the determinant of the Sylvester matrix and the \emph{discriminant} of a polynomial $p$ is the resultant of $p$ and its derivative. We study …

Counting real algebraic numbers with bounded derivative of minimal polynomial

In this paper we consider the problem of counting algebraic numbers $α$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_α(x)$ of $α$ is bounded, $|P_α'(α)| < Q^{1-v}$. This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of $α$ defined above on the interval $\left(-\frac12, \frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\frac{1}{7}v}$ for $Q>Q_0(n)$ and $1.4 \le v \le \frac{7}{16}(n+1)$. Our result is based on an improvement to the lemma on the order of zero approximation by…

多变量多项式的迭代 p 次幂循环结式的 p 进极限

The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials

本文研究多变量多项式迭代 p 次幂循环结式的 p 进收敛性，并将其应用于证明链环 Z_p^d-覆盖挠数的 p 进收敛，还以扭转 Whitehead 链环为例计算了极限，且证明在特定情形下该极限与 Kionke 引入的 p 进挠这一同伦不变量一致。

Heights of polynomials over lemniscates

We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and $p=\infty$ corresponds to the standard supremum norm. This special choice of the measure allows to find an explicit form for the geometric mean of a polynomial, and estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses, we establish explicit polynomials of lowest height, and also show their uniqueness. We discuss relations betwe…