Related papers: On parametric and generic polynomials with one par…

We compute new polynomials with Galois group $M_{11}$ over $\mathbb{Q}(t)$. These polynomials stem from various families of covers of $\mathbb{P}^1\mathbb{C}$ ramified over at least 4 points. Each of these families has features that make a…

We give a criterion whether given Eisenstein polynomials over a local field K define the same extension over K in terms of a certain non-Archimedean metric on the set of polynomials. The criterion and its proof depend on ramification…

A generalized Euler parameterization of a compact Lie group is a way for parameterizing the group starting from a maximal Lie subgroup, which allows a simple characterization of the range of parameters. In the present paper we consider the…

Let $L/K$ be a finite, totally ramified $p$-extension of complete local fields with residue fields of characteristic $p > 0$, and let $A$ be a $K$-algebra acting on $L$. We define the concept of an $A$-scaffold on $L$, thereby extending and…

We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of…

Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its…

We prove an effective form of Hilbert's irreducibility theorem for polynomials over a global field $K$. More precisely, we give effective bounds for the number of specializations $t\in \mathcal{O}_K$ that do not preserve the irreducibility…

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) \in K[z]$ be a separable polynomial of the form $z^\ell-c.$ Given $a \in K$, we examine the Galois groups and ramification groups…

Let~$E$ be a Hilbertian field of characteristic~$0$. R.W.K. Odoni conjectured that for every positive integer~$n$ there exists a polynomial~$f\in E[X]$ of degree~$n$ such that each iterate~$f^{\circ{k}}$ of~$f$ is irreducible and the Galois…

We generalize a result of F.\ Legrand about the existence of non-parametric Galois extensions for a given group $G$. More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension…

In this paper, we establish more identities of generalized multi poly-Euler polynomials with three parameters and obtain a kind of symmetrized generalization of the polynomials. Moreover, generalized multi poly-Bernoulli polynomials are…

The Concepts of poly-Bernoulli numbers $B_n^{(k)}$, poly-Bernoulli polynomials $B_n^{k}{(t)}$ and the generalized poly-bernoulli numbers $B_{n}^{(k)}(a,b)$ are generalized to $B_{n}^{(k)}(t,a,b,c)$ which is called the generalized…

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further…

We prove a parametric generalization of the classical Poincare-Perron theorem on stabilizing recurrence relations where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these…

Let $R$ be an integral domain over a field $k$, and $G$ a subgroup of the automorphism group of the polynomial ring $R[x_1,..., x_n]$ over $R$. In this paper, we discuss when $G$ is diagonalizable under the assumption that $G$ is…

If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional…

To a $2\times2$ matrix $G$ with complex entries, we relate the sequence of Laurent polynomial $L_n(z,G)=\tr \big(G\big[\begin{smallmatrix}z&0\\ 0&z^{-1}\end{smallmatrix}\big]G^{\ast}\big)^n$. It turns out that for each \(n\), the family…

This paper provides a realization of all classical and most exceptional finite groups of Lie type as Galois groups over function fields over F_q and derives explicit additive polynomials for the extensions. Our unified approach is based on…

Parahoric group schemes are certain possibly non-reductive, smooth, affine integral models of reductive group schemes defined over a henselian discretely valued field $K$ whose residue field is perfect. We show that any such group scheme…

Ideas and techniques from Khare's and Wintenberger's article on the proof of Serre's conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL_2(F_{l^r}) (for r running) occur as Galois…