Related papers: A necessary condition for solvability by radicals

Starting from the results of Charles Fefferman and Janos Koll\'ar in \texit{Continuous Solutions of Linear Equations} [1], we adopt a new approach based on Fefferman's techniques of Glaeser refinement to show a more general result than the…

We consider in detail an approach (proposed by the author earlier) where quantum states are described by elements of a linear space over a Galois field, and operators of physical quantities - by linear operators in this space. The notion of…

In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients.…

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending…

In this paper we consider the problem of Galois descent for suitably completed algebraic K-theory of fields. One of the main results is a suitable form of rigidity for Borel-style generalized equivariant cohomology with respect to certain…

In the mid sixties, A. Grothendieck envisioned a vast generalization of Galois theory to systems of polynomials in several variables, motivic Galois theory, and introduced tannakian categories on this occasion. In characteristic zero,…

We show that every linear algebraic group over an algebraically closed field of characteristic zero is the differential Galois group of a regular singular linear differential equation with rational function coefficients.

We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat.…

Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8) if p > 2 (resp. p = 2) characterizes its p-rationality. Then we give two short, very fast PARI Programs (\S \S 3.1, 3.2) testing if K (defined by an irreducible…

Using the framework of Colombeau algebras of generalized functions, we prove the existence and uniqueness results for global generalized solvability of semilinear hyperbolic systems with nonlinear nonlocal boundary conditions. We admit…

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…

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…

We classify general square systems of polynomial equations solvable in radicals. Expectedly, they are almost in a 1-to-1 correspondence with tuples of lattice polytopes of mixed volume not exceeding 4. The proof is based on the computation…

Classical approach of solvability problem has shed much light on what we can solve and what we cannot solve mathematically. Starting with quadratic equation, we know that we can solve it by the quadratic formula which uses square root.…

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently…

Let $L(X)$ be a monic $q$-linearized polynomial over $F_q$ of degree $q^n$, where $n$ is an odd prime. Recently Gow and McGuire showed that the Galois group of $L(X)/X-t$ over the field of rational functions $F_q(t)$ is $GL_n(q)$ unless…

Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x^4+px^2+qx+s with…

Let $\alpha$ be a non-zero algebraic number. Let $K$ be the Galois closure of $\mathbb{Q}(\alpha)$ with Galois group $G$ and $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. In this article, among the other results, we prove the…