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 = a n x n + · · · + a 1 x + a 0 ∈ Q [ x ]. For generic coefficients a n , . . . , a 0 , Gal( f ) ≃ S n . This means both 1 Gal( f/ Q ( a n , . . . , a 1 , a 0 )) ≃ S n , and 2 for a generic enough choice of coefficients in Q , Gal( f/ Q ) ≃ S n . 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 = ( f 1 , . . . , f n ) where each f i ∈ K [ x 1 , . . . , x n ], and define V ( F ) := V ( f 1 ) ∩ V ( f 2 ) ∩ · · · ∩ V ( f n ) n | f 1 ( x ) = · · · = f n ( x ) = 0 } . = { x ∈ K Assume the V ( f i ) intersect transversely, so that dim V ( F ) = 0. Define the splitting field Ω of F to be Ω := K ( z 1 , z 2 , . . . , z n | 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 = ( x 2 − 2 , y 3 − 2) over Q . We have √ √ 2 , ω i 3 V ( F ) = ( ± 2) for ω a 3rd root of unity and i ∈ { 0 , 1 , 2 } . The splitting field is √ √ 3 Ω = Q ( 2 , 2 , ω ) and Gal( F ) = Gal(Ω / Q ) ≃ C 2 × S 3 . A primitive element for Ω over Q √ √ 3 is given by 2 + 2 + ω , which has minimal polynomial g = x 12 + 6 x 11 + 9 x 10 − 18 x 9 − 48 x 8 + 6 x 7 + 17 x 6 + 510 x 5 + 1764 x 4 + 1350 x 3 + 573 x 2 − 642 x + 223 . The splitting field of g is Ω, and of course, Gal( g ) ≃ C 2 × S 3 . Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 4 / 20
Examples Consider the system F = ( x 2 − a, y 3 − b ) over K = Q ( a, b ), with √ V ( F ) = ( ±√ a, ω i 3 b ) . √ The splitting field is Ω = Q ( √ a, 3 b, ω ) and Gal( F ) ≃ C 2 × S 3 again. Had we taken K = C ( a, b ), the group would be C 2 × C 3 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 = ( x 2 − a, y 3 − b ), now over K = C ( a, b ). Let X be the variety cut out by F = 0 inside P 2 x,y × P 2 a,b . There is an obvious projection P 2 x,y × P 2 a,b → P 2 a,b onto the second component. Let π be its restriction to X . Geometrically, there is a finite branched cover of complex varieties π : X → P 2 a,b . and the monodromy group of π is C 2 × C 3 . 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 ( C 3 vs. S 3 ). Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 6 / 20
A note on specialization For F = ( x 2 − a, y 3 − 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 of 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 P 1 ( C ). Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 7 / 20
Polynomial systems with given support Any n -tuple a = ( a 1 , . . . , a n ) ∈ N n corresponds to the monomial x a = x a 1 1 . . . x a n n ∈ C [ x 1 , . . . , x n ] . In this way a finite set A 1 ⊆ N n corresponds to a space of polynomials c a x a : c a ∈ C C A 1 = � . a ∈ A 1 An n -tuple of supports A = ( A 1 , . . . , A n ) corresponds to C A = C A 1 ⊕ · · · ⊕ C A n , which is the space of all systems ( f 1 , . . . , f n ) of n polynomials in n variables with support A . Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 8 / 20
Mixed volume Let K 1 , . . . , K n be convex bodies in R n , equipped with Minkowski sum K 1 + K 2 = { x + y : x ∈ K 1 and y ∈ K 2 } . The mixed volume V ( K 1 , . . . , K n ) 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 ( K 1 , K 1 , . . . , K 1 ) = n !Vol( K 1 ) . For n = 2, V ( K 1 , K 2 ) = Vol( K 1 + K 2 ) − Vol( K 1 ) − Vol( K 2 ) , 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 = ( A 1 , . . . , A n ) and let F = ( f 1 , . . . , f n ) ∈ C A . 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 B A ⊆ C A such that, for all F ∈ C A \ B A , the number of isolated solutions in ( C ∗ ) n of F ( x ) = 0 equals the mixed volume of the convex hulls of the A i . Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 10 / 20
The Bernstein-Kouchnirenko theorem The system a 0 + a 1 x + a 2 y + a 3 xy = 0 b 0 + b 1 x 2 y + b 2 xy 2 = 0 generically has 4 solutions in ( C ∗ ) 2 . It corresponds to the tuples A 1 = { (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1) } A 2 = { (0 , 0) , (2 , 1) , (1 , 2) } Using our formula for mixed volume in dimension 2, it is easy to show that V ( N 1 , N 2 ) = 4, where N i is the convex hull of A i . 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 = ( A 1 , . . . , A n ) be a prescribed support. We say A is reduced if the A i do not lie in a proper sublattice of Z n and irreducible if no k of the A i lie in a k -dimensional sublattice of Z n , up to translating any number of the A i . A is non-reduced iff there is a non-invertible map ψ : Z n → Z n such that A ⊆ ψ ( Z n ), after translation if necessary. Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 12 / 20
Galois group of a polynomial system Let A = ( A 1 , . . . , A n ) be a support and let F ∈ C A . Let V be the mixed volume of the convex hulls of the A i . Theorem (Esterov) 1 If A is reduced and irreducible, then Gal ( F ) ≃ S V . 2 If Gal ( F ) < S V , 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 f i 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 : Z n → Z n and B ⊆ Z n 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 A 1 = A 2 = · · · = A n . 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 a i x i ∈ C ( { a i : i ∈ A } )[ x ] . � f ( x ) = i ∈ A with deg f = n . Let d = gcd( A ), so f decomposes as f = g ◦ x d and g has no further decomposition. Theorem (Esterov-Lang, B.-Sottile) Gal ( f ) ≃ C d ≀ S n/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 ◦ x d , and let z 1 , . . . , z k be the roots of g . The roots of f break up into k blocks of size d . The i th block consists of the roots of x d − z i , and Gal( x d − z i ) ≃ C d for each i . The group C d ≀ S k 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 ) ≃ C d ≀ S k , we use specializations of f to C ( t ). Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 16 / 20
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