Roots of Chebyshev Polynomials: a purely algebraic approach

第二类广义切比雪夫多项式的算术例外性

Arithmetic exceptionality of generalized Chebyshev polynomials of the second kind

本文通过研究有限域参数化的分圆元素范数，证明与根系$A_2$相关的第二类广义切比雪夫多项式不具有算术例外性。

Discriminants of Chebyshev Radical Extensions

Let t be any integer and fix an odd prime ell. Let Phi(x) = T_ell^n(x)-t denote the n-fold composition of the Chebyshev polynomial of degree ell shifted by t. If this polynomial is irreducible, let K = bbq(theta), where theta is a root of Phi. A theorem of Dedekind's gives a condition on t for which K is monogenic. For other values of t, we apply the Montes algorithm to obtain a formula for the discriminant of K and to compute basis elements for the ring of integers O_K.

关于二次无理数的高效逼近

On efficient approximation of quadratic irrationals, signed Chebyshev polynomials and Householder's method

本文提出高效计算二次无理数收敛的算法，证明在拉格朗日定理的伽罗瓦细化条件下，平方根的收敛序列的某些抽取是带符号的切比雪夫序列，并可通过Householder方法生成。

Distribution asymptotique des valeurs propres des endomorphismes de Frobenius [d'après Abel, Chebyshev, Robinson, ...]

We consider unitary polynomials $P \in Z[X]$ whose roots $(x_i)$ belong to a given compact $K$ of $C$. To such a polynomial we associate the measure $μ_P$ on $K$ which is the mean value of the Dirac measures $δ_{x_i}$. What are the limits of the measures $μ_P$ when $P$ varies ? In particular, what are their supports? We give partial answers to such questions, especially when $K$ is contained in $R$.

Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials

Square roots $s$ of sums of $M$ consecutive integer squares starting from $a^{2}\geq1$ are integers if $M\equiv0,9,24$ or $33(mod\,72)$; or $M\equiv1,2$ or $16(mod\,24)$; or $M\equiv11(mod\,12)$ and cannot be integers if $M\equiv3,5,6,7,8$ or $10(mod\,12)$. Finding all solutions with $s$ integer requires to solve a Diophantine quadratic equation in variables $a$ and $s$ with $M$ as a parameter. If $M$ is not a square integer, the Diophantine quadratic equation in variables $a$ and $s$ is transformed into a generalized Pell equation whose form depends on the $M(mod\,4)$ congruent value, and who…

Polynomials with integer coefficients and their zeros

We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.