Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.

Explicit determination of generalized symmetric and alternating Galois groups

Abstract: This paper shows a probabilistic algorithm to decide whether the Galois group of a given irreducible polynomial with rational coefficients is the generalized symmetric group Cp o Sm or the generalized alternating group Cp o Am. In the affirmative case, we give generators of the group with their action on the set of roots of the polynomial.

Note sur les corps 2-rationnels

Abstract: We compute the Galois group of the maximal 2-ramified pro-2-extension of a 2-rational number field

Horizontal gradient of polynomial functions for the standard Engel structure on $\mathbb R^4$

Abstract: We investigate the set $V_f$ of horizontal critical points of a polynomial function $f$ for the standard Engel structure defined by the 1-forms $\omega_3=dx_3-x_1dx_2,$ $\omega_4=dx_4-x_3dx_2$, endowed with the sub-Riemannian metric $g_{SR}=dx_1^2+dx_2^2$. For a generic polynomial, we show that the intersection of any fiber of $f$ and $V_f$ does not contain a horizontal curve. Then we prove that each trajectory of the horizontal gradient of $f$ approaching the set $V_f$ has a limit.

The $p$-adic Riemann Hypothesis For Expnonential Sums.

Abstract: The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is defined. A $p$-adic Riemann hypothesis is formulated. It asserts that the Newton polygon of the $L$-function coincides with the Frobenius polygon when $p$ is large enough. This $p$-adic Riemann hypothesis is proved when $n=2$ and $p\equiv-1({\rm mod }\ d)$. In general, it is proved that the Newton polygon of the $L$-function lies above the Frobenius polygon with coincide endpoints when $p$ is large enough.