Galois theory in several variables: a number theory perspective - - PowerPoint PPT Presentation

Galois theory in several variables: a number theory perspective

Andrew Bridy (joint with Frank Sottile)

Yale University

September 1, 2020

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 1 / 20

A classic example

Let f = anxn + · · · + a1x + a0 ∈ Q[x]. For generic coefficients an, . . . , a0, Gal(f) ≃ Sn. This means both

1 Gal(f/Q(an, . . . , a1, a0)) ≃ Sn, and 2 for a generic enough choice of coefficients in Q, Gal(f/Q) ≃ Sn.

The second claim follows from the first by Hilbert’s Irreducibility Theorem (but defining “generic enough” is not always easy.) What happens in several variables?

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 2 / 20

Galois group of a polynomial system

Let K be a characteristic 0 field with algebraic closure K. Let F = (f1, . . . , fn) where each fi ∈ K[x1, . . . , xn], and define V (F) := V (f1) ∩ V (f2) ∩ · · · ∩ V (fn) = {x ∈ K

n | f1(x) = · · · = fn(x) = 0}.

Assume the V (fi) intersect transversely, so that dim V (F) = 0. Define the splitting field Ω of F to be Ω := K(z1, z2, . . . , zn | z ∈ V (F)), that is, Ω contains all coordinates of points in V (F). Define Gal(F) := Gal(Ω/K). Exercise: Ω is a Galois extension of K, i.e., the splitting field of a single univariate polynomial.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 3 / 20

Examples

We compute the Galois group of F = (x2 − 2, y3 − 2) over Q. We have V (F) = (± √ 2, ωi 3 √ 2) for ω a 3rd root of unity and i ∈ {0, 1, 2}. The splitting field is Ω = Q( √ 2,

3

√ 2, ω) and Gal(F) = Gal(Ω/Q) ≃ C2 × S3. A primitive element for Ω over Q is given by √ 2 +

3

√ 2 + ω, which has minimal polynomial g = x12 + 6x11 + 9x10 − 18x9 − 48x8 + 6x7 + 17x6 + 510x5 + 1764x4 + 1350x3 + 573x2 − 642x + 223. The splitting field of g is Ω, and of course, Gal(g) ≃ C2 × S3.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 4 / 20

Examples

Consider the system F = (x2 − a, y3 − b) over K = Q(a, b), with V (F) = (±√a, ωi 3 √ b). The splitting field is Ω = Q(√a,

3

√ b, ω) and Gal(F) ≃ C2 × S3 again. Had we taken K = C(a, b), the group would be C2 × C3 instead, because C already contains a 3rd root of unity.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 5 / 20

Examples

Consider again the system F = (x2 − a, y3 − b), now over K = C(a, b). Let X be the variety cut out by F = 0 inside P2

x,y × P2 a,b. There is an

• bvious projection

P2

x,y × P2 a,b → P2 a,b

• nto the second component. Let π be its restriction to X.

Geometrically, there is a finite branched cover of complex varieties π : X → P2

a,b.

and the monodromy group of π is C2 × C3. The monodromy group equals the Galois group of the function field extension K(X)/C(a, b) (Harris). This geometric picture can be invoked over Q, but it produces a constant field extension (C3 vs. S3).

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 6 / 20

A note on specialization

For F = (x2 − a, y3 − b), the Galois group over Q(a, b) is preserved if we specialize the coefficients to some a ∈ Q and b ∈ Q and take the Galois group over Q, for “most” specializations. This is a consequence

• f Hilbert’s Irreducibility Theorem.

If we take the Galois group over C(a, b), we cannot hope to specialize to a ∈ C, b ∈ C and preserve the Galois group. However, we can introduce a parameter t and specialize to a, b ∈ C(t), again preserving the group under most specializations. Galois extensions K/Q are analogous to Galois extensions K/C(t), which correspond to Galois covers of P1(C).

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 7 / 20

Polynomial systems with given support

Any n-tuple a = (a1, . . . , an) ∈ Nn corresponds to the monomial xa = xa1

1 . . . xan n ∈ C[x1, . . . , xn].

In this way a finite set A1 ⊆ Nn corresponds to a space of polynomials CA1 =   

• a∈A1

caxa : ca ∈ C    . An n-tuple of supports A = (A1, . . . , An) corresponds to CA = CA1 ⊕ · · · ⊕ CAn, which is the space of all systems (f1, . . . , fn) of n polynomials in n variables with support A.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 8 / 20

Mixed volume

Let K1, . . . , Kn be convex bodies in Rn, equipped with Minkowski sum K1 + K2 = {x + y : x ∈ K1 and y ∈ K2}. The mixed volume V (K1, . . . , Kn) is the unique real-valued function on n-tuples of convex bodies that is symmetric, multilinear with respect to Minkowski sum, and normalized so that V (K1, K1, . . . , K1) = n!Vol(K1). For n = 2, V (K1, K2) = Vol(K1 + K2) − Vol(K1) − Vol(K2), and this formula extends appropriately to n ≥ 3.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 9 / 20

The Bernstein-Kouchnirenko theorem

Let A = (A1, . . . , An) and let F = (f1, . . . , fn) ∈ CA. The Bernstein-Kouchnirenko theorem predicts the generic number of solutions of the system F(x) = 0 in (C∗)n.

Theorem (Bernstein-Kouchnirenko)

There is an algebraic set BA ⊆ CA such that, for all F ∈ CA \ BA, the number of isolated solutions in (C∗)n of F(x) = 0 equals the mixed volume of the convex hulls of the Ai.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 10 / 20

The Bernstein-Kouchnirenko theorem

The system a0 + a1x + a2y + a3xy = 0 b0 + b1x2y + b2xy2 = 0 generically has 4 solutions in (C∗)2. It corresponds to the tuples A1 = {(0, 0), (0, 1), (1, 0), (1, 1)} A2 = {(0, 0), (2, 1), (1, 2)} Using our formula for mixed volume in dimension 2, it is easy to show that V (N1, N2) = 4, where Ni is the convex hull of Ai. B´ ezout’s Theorem predicts 6 solutions, 2 of which are generically at ∞.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 11 / 20

Reduced and irreducible supports

Let A = (A1, . . . , An) be a prescribed support. We say A is reduced if the Ai do not lie in a proper sublattice of Zn and irreducible if no k of the Ai lie in a k-dimensional sublattice of Zn, up to translating any number of the Ai. A is non-reduced iff there is a non-invertible map ψ : Zn → Zn such that A ⊆ ψ(Zn), after translation if necessary.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 12 / 20

Galois group of a polynomial system

Let A = (A1, . . . , An) be a support and let F ∈ CA. Let V be the mixed volume of the convex hulls of the Ai.

Theorem (Esterov)

1 If A is reduced and irreducible, then Gal(F) ≃ SV . 2 If Gal(F) < SV , then Gal(F) is imprimitive.

In Esterov’s setting, Gal(F) is a monodromy group. We can also take it to be a Galois group as we have described, with the coefficients of the fi as indeterminates.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 13 / 20

The non-reduced case

Suppose A is non-reduced with corresponding polynomial system F. We may choose ψ : Zn → Zn and B ⊆ Zn so that A = ψ(B) for B

• reduced. Let G be the system with support B. There is an embedding

Gal(F) ֒ → coker ψ ≀ Gal(G).

Conjecture (Esterov)

If A is irreducible, this embedding is an isomorphism. Unfortunately, this conjecture is false in general, though true if n = 1. See Esterov-Lang for a complete accounting in the case A1 = A2 = · · · = An. In full generality, not much is known.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 14 / 20

The univariate non-reduced case

Let f(x) =

• i∈A

aixi ∈ C({ai : i ∈ A})[x]. with deg f = n. Let d = gcd(A), so f decomposes as f = g ◦ xd and g has no further decomposition.

Theorem (Esterov-Lang, B.-Sottile)

Gal(f) ≃ Cd ≀ Sn/d Esterov-Lang prove this as a special case of their general argument, which uses toric geometry and topology. Bridy-Sottile’s argument is purely algebraic, and should to some extent carry over to positive characteristic.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 15 / 20

The univariate non-reduced case

Let f = g ◦ xd, and let z1, . . . , zk be the roots of g. The roots of f break up into k blocks of size d. The ith block consists

• f the roots of xd − zi, and

Gal(xd − zi) ≃ Cd for each i. The group Cd ≀ Sk consists of all permutations of the kd roots of f that respect the block structure and act cyclically within each block, so it is the largest possible Galois group of f. To show Gal(f) ≃ Cd ≀ Sk, we use specializations of f to C(t).

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 16 / 20

The univariate non-reduced case

Let L be the splitting field of g, so Gal(L) ≃ Sk. Let Mi be the splitting field of xd = zi, and let M be the splitting field of f, so that M = M1M2 · · · Mk. The hard part of the argument is to show that the Mi are disjoint over L, in the sense that Mi ∩

• j=i

Mj = L. This is done by specializing to find primes pi of L that ramify in Mi and not in any other Mj. With this at hand, Gal(M/L) ≃

k

• i=1

Gal(Mi/L) ≃ Cd × . . . Cd, which finishes the argument.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 17 / 20

The univariate non-reduced case

This is the diagram of fields for d = 3 and deg g = 4. The generic Galois group C3 ≀ S4 acts by automorphisms on this tree.

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 18 / 20

Further questions

How much carries over to positive characteristic? Can we prove probabilistic statements for Galois groups of random polynomial systems over Q? How does Gal(F) relate to splitting and ramification of primes in several variables? Is there a Dedekind-Kummer theorem or a Chebotarev density theorem about factorizations mod p? What is the correct formulation of the inverse Galois problem for systems of complex polynomials with generic coefficients? Is the multivariate approach useful for attacking univariate Galois problems by adding more variables?

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 19 / 20

Galois theory in several variables: a number theory perspective

Andrew Bridy (joint with Frank Sottile)

Yale University

September 1, 2020

Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 20 / 20