scholarly journals Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
A. Esterov ◽  
L. Lang
Keyword(s):  
Galois Group ◽  
Wreath Product ◽  
Galois Theory ◽  
General System ◽  
Wreath Products ◽  
Galois Groups ◽  
Complex Polynomial ◽  
Polynomial Equations ◽  
Sparse Polynomial ◽  
System Of Polynomial Equations

AbstractWe introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial $$f(x) = c_0 + c_1 x^{d_1} + \cdots + c_k x^{d_k}$$ f ( x ) = c 0 + c 1 x d 1 + ⋯ + c k x d k by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable $$y=x^d$$ y = x d , and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over $$d_k/d$$ d k / d elements and $${\mathbb {Z}}/d{\mathbb {Z}}$$ Z / d Z . We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.

scholarly journals THE DISTRIBUTION OF NUMBER FIELDS WITH WREATH PRODUCTS AS GALOIS GROUPS

2012 ◽  
Vol 08 (03) ◽  
pp. 845-858 ◽  
Author(s):  
JÜRGEN KLÜNERS
Keyword(s):  
Asymptotic Behavior ◽  
Galois Group ◽  
Wreath Product ◽  
Symmetric Groups ◽  
Number Fields ◽  
Wreath Products ◽  
Constant Factor ◽  
Galois Groups ◽  
Mild Conditions ◽  
Counting Functions

Let G be a wreath product of the form C2 ≀ H, where C2 is the cyclic group of order 2. Under mild conditions for H we determine the asymptotic behavior of the counting functions for number fields K/k with Galois group G and bounded discriminant. Those counting functions grow linearly with the norm of the discriminant and this result coincides with a conjecture of Malle. Up to a constant factor these groups have the same asymptotic behavior as the conjectured one for symmetric groups.

scholarly journals Galois theory for general systems of polynomial equations

2019 ◽  
Vol 155 (2) ◽  
pp. 229-245 ◽  
Author(s):  
A. Esterov
Keyword(s):  
Galois Theory ◽  
General System ◽  
Mixed Volume ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Change Of Variables ◽  
General Polynomial ◽  
General Systems ◽  
Systems Of Polynomial Equations

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

scholarly journals Computation of Galois groups of rational polynomials

2014 ◽  
Vol 17 (1) ◽  
pp. 141-158 ◽  
Author(s):  
Claus Fieker ◽  
Jürgen Klüners
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Large Degree ◽  
Rational Numbers ◽  
Galois Groups ◽  
Multivariate Polynomial ◽  
Theoretical Methods ◽  
Algorithmic Solution ◽  
Rational Polynomials ◽  
General Method

AbstractComputational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups $H<G$, a $G$-relative $H$-invariant, that is a multivariate polynomial $F$ that is $H$-invariant, but not $G$-invariant. While generic, theoretical methods are known to find such $F$, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.

Galois theory of Salem polynomials

2009 ◽  
Vol 148 (1) ◽  
pp. 47-54 ◽  
Author(s):  
CHRISTOS CHRISTOPOULOS ◽  
JAMES MCKEE
Keyword(s):  
Unit Circle ◽  
Galois Group ◽  
Galois Theory ◽  
Minimal Polynomial ◽  
Galois Groups ◽  
Cyclotomic Polynomials ◽  
Salem Number ◽  
Extra Structure

AbstractLet f(x) ∈ [x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1/r1, r2, 1/r2, . . ., rd, 1/rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 + 1/r1, . . ., rd + 1/rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg ≅ Gf/N, where N is isomorphic to a subgroup of C2d. In a naive sense, the generic case is Gf ≅ C2d ⋊ Sd, with N ≅ C2d and Gg ≅ Sd. When f(x) has extra structure this may be reflected in the Galois group, and it is not always true even that Gf ≅ N ⋊ Gg. For example, for cyclotomic polynomials f(x) = Φn(x) it is known that Gf ≅ N ⋊ Gg if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4.In this paper we deal with irreducible reciprocal monic polynomials f(x) ∈ [x] that are ‘close’ to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f(x) has degree at least 4, this means that f(x) is the minimal polynomial of a Salem number. We show that in this case one always has Gf ≅ N ⋊ Gg, and moreover that N ≅ C2d or C2d−1, with the latter only possible if d is odd.

Effective Galois theory

1981 ◽  
Vol 46 (2) ◽  
pp. 385-392 ◽  
Author(s):  
Peter La Roche
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Galois Groups ◽  
Profinite Group ◽  
Classical Literature ◽  
Profinite Groups ◽  
Algebraic Numbers ◽  
Recursively Enumerable ◽  
Free Profinite Group ◽  
Infinite Degree

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups (i.e. totally disconnected, compact, Hausdorff groups). Metakides and Nerode [7] produced two recursively presented algebraic extensionsK⊂Fof the rationals such thatFis abelian,Fis of infinite degree overK, and the Galois group ofFoverK, although of cardinalityc, has only one recursive element (viz. the identity). This indicated the limits of effectiveness for Krull's theory. (The Galois theory offiniteextensions is completely effective.) Nerode suggested developing a natural effective version of Krull's theory (done here in §1).It is evident from the classical literature that the free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield (done here in §2). The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield.

scholarly journals A note on Galois groups of algebraic closures

1976 ◽  
Vol 21 (1) ◽  
pp. 12-15 ◽  
Author(s):  
Don Krakowski
Keyword(s):  
Galois Group ◽  
Fundamental Theorem ◽  
Inverse Limit ◽  
Galois Theory ◽  
Algebraic Closure ◽  
Galois Groups ◽  
Point Of View ◽  
Roots Of Unity ◽  
Profinite Groups ◽  
Profinite Topology

A group will be called full if it is the Galois group of an algebraic closure of a field. In this paper we first investigate full Abelian groups and classify them. Then we examine full groups from the point of view of how we can operate on them and still maintain the property of being full. Of course, by the fundamental theorem of the infinite Galois theory, closed subgroups (with standard profinite topology) of full groups are full. In general, products of full groups are notfull (for example, Z2 × Z2 is not full, by a theorem of Artin and Schreier (1927)); however we produce a set of groups which can always be attached as direct factors to full groups and still retain full groups. For the definition and basic properties of profinite groups we refer the reader to Cassels and Frohlich (1967). It has been shown by Leptin (1955) and independently by the author that profinite groups are Galois groups. The author (1971) has shown that if G is profinite, then G = Gdl(k/L) where L may have any desired characteristic and contains all primitive nth roots of unity, for all n. We will denote by p the pro-p-group which is the inverse limit of cyclic p-groups. (This is the group of p-adic integers).

A galois theory for the banach algebra of continuous symmetric functions on absolute galois groups

2004 ◽  
Vol 45 (3-4) ◽  
pp. 349-358 ◽  
Author(s):  
Angel Popescu ◽  
Nicolae Popescu ◽  
Alexandru Zaharescu
Keyword(s):  
Banach Algebra ◽  
Symmetric Functions ◽  
Galois Theory ◽  
Galois Groups

scholarly journals WREATH PRODUCTS, NILPOTENT ORBITS AND SYMPLECTIC DEFORMATIONS

2007 ◽  
Vol 18 (05) ◽  
pp. 473-481
Author(s):  
BAOHUA FU
Keyword(s):  
Wreath Product ◽  
Wreath Products ◽  
Nilpotent Orbit ◽  
Nilpotent Orbits ◽  
Symplectic Resolutions

We recover the wreath product X ≔ Sym 2(ℂ2/± 1) as a transversal slice to a nilpotent orbit in 𝔰𝔭6. By using deformations of Springer resolutions, we construct a symplectic deformation of symplectic resolutions of X.

scholarly journals A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

2009 ◽  
Vol 47 (5) ◽  
pp. 3608-3623 ◽  
Author(s):  
Daniel J. Bates ◽  
Jonathan D. Hauenstein ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Solution Set ◽  
Polynomial Equations ◽  
Local Dimension ◽  
System Of Polynomial Equations

scholarly journals On the Galois group of three classes of trinomials

2021 ◽  
Vol 7 (1) ◽  
pp. 212-224
Author(s):  
Lingfeng Ao ◽  
◽  
Shuanglin Fei ◽  
Shaofang Hong
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Prime Number ◽  
Galois Groups

<abstract><p>Let $ n\ge 8 $ be an integer and let $ p $ be a prime number satisfying $ \frac{n}{2} &lt; p &lt; n-2 $. In this paper, we prove that the Galois groups of the trinomials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ T_{n, p, k}(x): = x^n+n^kp^{(n-1-p)k}x^p+n^kp^{nk}, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ S_{n, p}(x): = x^n+p^{n(n-1-p)}n^px^p+n^pp^{n^2} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ E_{n, p}(x): = x^n+pnx^{n-p}+pn^2 $\end{document} </tex-math></disp-formula></p> <p>are the full symmetric group $ S_n $ under several conditions. This extends the Cohen-Movahhedi-Salinier theorem on the irreducible trinomials $ f(x) = x^n+ax^s+b $ with integral coefficients.</p></abstract>

Sign in / Sign up

Export Citation Format

Share Document