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Galois Theory in Several Variables: a Number Theory Perspective

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Galois Theory in Several Variables: a Number Theory Perspective 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 n Let f = anx + ··· + a1x + a0 2 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 2 K[x1; : : : ; xn], and define V (F ) := V (f1) \ V (f2) \···\ V (fn) n = fx 2 K j f1(x) = ··· = fn(x) = 0g: 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 j z 2 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 2 3 We compute the Galois group of F = (x − 2; y − 2) over Q. We have p p V (F ) = (± 2;!i 3 2) for ! a 3rd root of unity and i 2 f0; 1; 2g. The splitting field is p p3 Ω = Q( 2; 2;!) and Gal(F )p = Gal(Ωp =Q) ' C2 × S3. A primitive element for Ω over Q is given by 2 + 3 2 + !, 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 2 3 Consider the system F = (x − a; y − b) over K = Q(a; b), with p p V (F ) = (± a; !i 3 b): p p3 The splitting field is Ω = Q( a; b; !) 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 2 3 Consider again the system F = (x − a; y − b), now over K = C(a; b). 2 2 Let X be the variety cut out by F = 0 inside Px;y × Pa;b. There is an obvious projection 2 2 2 Px;y × Pa;b ! Pa;b onto the second component. Let π be its restriction to X. Geometrically, there is a finite branched cover of complex varieties 2 π : X ! Pa;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 2 3 For F = (x − a; y − b), the Galois group over Q(a; b) is preserved if we specialize the coefficients to some a 2 Q and b 2 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 2 C, b 2 C and preserve the Galois group. However, we can introduce a parameter t and specialize to a; b 2 C(t), again preserving the group under most specializations. Galois extensions K=Q are analogous to Galois extensions K=C(t), 1 which correspond to Galois covers of P (C). Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 7 / 20 Polynomial systems with given support n Any n-tuple a = (a1; : : : ; an) 2 N corresponds to the monomial a a1 an x = x1 : : : xn 2 C[x1; : : : ; xn]: n In this way a finite set A1 ⊆ N corresponds to a space of polynomials 8 9 < = A1 X a C = cax : ca 2 C : :a2A1 ; An n-tuple of supports A = (A1;:::;An) corresponds to A A A C = C 1 ⊕ · · · ⊕ C n ; 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 n Let K1;:::;Kn be convex bodies in R , equipped with Minkowski sum K1 + K2 = fx + y : x 2 K1 and y 2 K2g: 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 A Let A = (A1;:::;An) and let F = (f1; : : : ; fn) 2 C . The Bernstein-Kouchnirenko theorem predicts the generic number of ∗ n solutions of the system F (x) = 0 in (C ) . Theorem (Bernstein-Kouchnirenko) A A There is an algebraic set BA ⊆ C such that, for all F 2 C n BA, the ∗ n number of isolated solutions in (C ) 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 2 2 b0 + b1x y + b2xy = 0 ∗ 2 generically has 4 solutions in (C ) . It corresponds to the tuples A1 = f(0; 0); (0; 1); (1; 0); (1; 1)g A2 = f(0; 0); (2; 1); (1; 2)g 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'sTheorem predicts 6 solutions, 2 of which are generically at 1. 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 n reduced if the Ai do not lie in a proper sublattice of Z and n irreducible if no k of the Ai lie in a k-dimensional sublattice of Z , up to translating any number of the Ai. n n A is non-reduced iff there is a non-invertible map : Z ! Z such n that A ⊆ (Z ), after translation if necessary. Andrew Bridy (Yale University) Multivariate Galois theory September 1, 2020 12 / 20 Galois group of a polynomial system A Let A = (A1;:::;An) be a support and let F 2 C . 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 . n n n We may choose : Z ! Z and B ⊆ Z so that A = (B) for B reduced. Let G be the system with support B. There is an embedding Gal(F ) ,! coker o 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 X i f(x) = aix 2 C(fai : i 2 Ag)[x]: i2A 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 o 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 d Let f = g ◦ x , 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 d of the roots of x − zi, and d Gal(x − zi) ' Cd for each i. The group Cd o 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 o 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.
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