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The Class of Eisenbud–Khimshiashvili–Levine Is the Local A1-Brouwer Degree

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The Class of Eisenbud–Khimshiashvili–Levine Is the Local A1-Brouwer Degree THE CLASS OF EISENBUD–KHIMSHIASHVILI–LEVINE IS THE LOCAL A1-BROUWER DEGREE JESSE LEO KASS AND KIRSTEN WICKELGREN ABSTRACT. Given a polynomial function with an isolated zero at the origin, we prove that the local A1-Brouwer degree equals the Eisenbud–Khimshiashvili–Levine class. This an- swers a question posed by David Eisenbud in 1978. We give an application to counting nodes together with associated arithmetic information by enriching Milnor’s equality be- tween the local degree of the gradient and the number of nodes into which a hypersurface singularity bifurcates to an equality in the Grothendieck–Witt group. We prove that the Eisenbud–Khimshiashvili–Levine class of a polynomial function with an isolated zero at the origin is the local A1-Brouwer degree, a result that answers a question of Eisenbud. n n The classical local Brouwer degree deg0(f) of a continuous function f: R R with an isolated zero at the origin is the image deg(f=jfj) 2 Z of the map of (n - 1)-spheres n-1 n-1 ! f=jfj: S S ; > 0 sufficiently small, n-1 n-1 under the global Brouwer degree homomorphism deg:[S ;S ] Z. ! When f is a C function, Eisenbud–Levine and independently Khimshiashvili con- ! structed a real nondegenerate1 symmetric bilinear form (more precisely, an isomorphism n class of such forms) w0(f) on the local algebra Q0(f) := C0 (R )=(f) and proved 1 (1) deg0(f) = the signature of w0(f) ([EL77, Theorem 1.2], [Khi77]; see also [AGZV12, Chapter 5] and [Khi01]). If we further n n assume that f is real analytic, then we can form the complexification fC : C C , and Palamodov [Pal67, Corollary 4] proved an analogous result for fC: ! (2) deg0(fC) = the rank of w0(f): Eisenbud observed that the definition of w0(f) remains valid when f is a polynomial with coefficients in an arbitrary field k and asked whether this form can be identified with a degree in algebraic topology [Eis78, Some remaining questions (3)]. Here we answer 1 Eisenbud’s question by proving that w0(f) is the local Brouwer degree in A -homotopy theory. More specifically, we prove n n Main Theorem. If f: Ak Ak has an isolated zero at the origin, then A1 (3) deg (f) = the stable isomorphism class of w0(f). 0! Date: August 12, 2021. 2010 Mathematics Subject Classification. Primary 14F42; Secondary 14B05, 55M25. 1 Morel described the degree map in A1-homotopy theory in his 2006 presentation at the International Congress of Mathematicians [Mor06]. In A1-homotopy theory, one of n n-1 several objects that plays the role of the sphere is Pk =Pk , the quotient of n-dimensional projective space by the (n-1)-dimensional projective space at infinity. Morel constructed a group homomorphism A1 n n-1 n n-1 deg :[Pk =Pk ; Pk =Pk ] GW(k) 1 n n-1 from the A -homotopy classes of endomorphisms of Pk =Pk to the Groethendieck–Witt group, which is the groupification of the monoid of (isomorphism! classes of) nondegen- erate symmetric bilinear forms over k. The local degree is defined in terms of the global degree in the natural manner, as we explain in Section 2. The proof of the Main Theorem runs as follows. When f has a simple zero at the origin, we prove the result by directly computing that both sides of (3) are represented by the class of the Jacobian hdet( @fi (0))i. When f has a simple zero at a nonrational point, we @xj show an analogous equality using work of Hoyois [Hoy14]. Using the result for a simple zero, we then prove the result when f has an arbitrary zero. We begin by reducing to n n the case where f is the restriction of a morphism F: Pk Pk satisfying certain technical conditions (those in Assumption 19) that include the condition that all zeros of F other n than the origin are simple. For every closed point x 2 Ak!, Scheja–Storch have constructed a bilinear form whose class wx(F) equals the Eisenbud–Khimshiashvili–Levine class when x = 0. From the result on simple zeros, we deduce that (4) degx(F) = wx(F) -1 -1 x2Xf (y) x2Xf (y) n holds for y 2 Ak (k) a regular value. For y arbitrary (and possibly not a regular value), we show that both sums in (4) are independent of y, allowing us to conclude that (4) holds for all y. In particular, equality holds for y = 0. For y = 0, we have degx(F) = wx(F) for x 2 f-1(0) not equal to the origin by the result for simple zeros, and taking differences, we deduce the equality for deg0(F) = w0(F), which is the Main Theorem. We propose counting singularities arithmetically, and in Section 6, we do so using the Main Theorem in the manner that we now describe. Suppose char k 6= 2 and n is even, and let f 2 k[x1; : : : ; xn] be the equation of an isolated hypersurface singularity n 0 2 X := ff = 0g ⊂ Ak 1 A1 A1 at the origin. We define the arithmetic (or A -) Milnor number by µ (f) := deg0 (grad(f)) and show that this invariant is an arithmetic count of the nodes (or A1-singularities) to which X bifurcates. Suppose grad(f) is finite and separable. Then for general (a1; : : : ; an) 2 n Ak (k), the family (5) f(x1; : : : ; xn) + a1x1 + ··· + anxn = t over the affine t-line contains only nodal fibers as singular fibers. For simplicity, assume that the origin is the only zero of grad f and the nodes appearing in (5) all have residue field k (rather than a nontrivial extension). We then have (6) A1 2 2 µ (f) = #(nodes with henselization fu1x1 + ··· + unxn = 0g) · hu1 : : : uni in GW(k). X 2 2 Here the sum runs over isomorphic classes of henselizations of rings k[x1; : : : ; xn]=u1x1 + 2 ··· + unxn, and the naive count is of nodal fibers of (5). Taking the rank of both sides of Equation (6), we deduce that the number of nodal 1 fibers equals the rank of µA (f). When k = C, this fact was observed by Milnor [Mil68, page 113, Remark], and (6) should be viewed as an enrichment of Milnor’s result from an equality of integers to an equality of classes in GW(k). When k = R, the real realization of Equation (6) was essentially proven by Wall [Wal83, page 347, esp. second displayed equation]. Through Equation (6), the arithmetic Milnor number provides a computable constraint on the nodes to which a hypersurface singularity can bifurcate. As an illustration, con- 2 3 sider the cusp (or A2-singularity) fx1 + x2 = 0g over the field Qp of p-adic numbers. A A1 ∗ ∗ 2 computation shows that µ (f) has rank 2 and discriminant -1 2 Qp=(Qp) . When p = 5, ∗ ∗ 2 we have -1 6= 1 · 2 in Qp=(Qp) , so we conclude that the cusp cannot bifurcate to the split 2 2 2 A1 node fx1 + x2 = 0g and the nonsplit node fx1 + 2 · x2 = 0g. When p = 11, µ (f) does not provide such an obstruction and in fact, those two nodes are the singulars fibers of 2 3 x1 + x2 + 10 · x2 = t. We discuss this example in more detail towards the end of Section 6. Arithmetic Milnor numbers, and other local A1-degrees, also appear in enumerative results of M. Levine. (Note: this person is different from the second author of [EL77]). Indeed, in [Lev17] Levine establishes a formula giving an enumerative count of the sin- gular fibers of a suitable fibration of a smooth projective variety over a curve in which an 1 isolated singularity of a fiber is weighted by µA (f). Levine’s formula should be viewed 1 as a global analogue of (6). Levine also computes µA (f) when f satisfies a certain “diago- nizability” hypothesis. A different application of the A1-degree to enumerative geometry is given by the present authors in [KW17]. There the authors study a weighted count of the lines on a cubic surface with the weights defined as local A1-degrees. We discuss the application to cubic surfaces in Section 7. The results of this paper are related to results in the literature. We have already dis- cussed the work of Eisenbud–Khimshiashvili–Levine and Palamodov describing w0(f) when k = R; C. When k is an ordered field, Bottger–Storch¨ studied the properties of n n w0(f) in [BS11]. They defined the mapping degree of f: Ak Ak to be the signature of w0(f) [BS11, 4.2 Definition, 4.3 Remark] and then proved that the mapping degree is a signed count of the points in the preimage of a regular value! [BS11, Theorem 4.5]. 0 Grigor ev–Ivanov studied w0(f) when k is an arbitrary field in [GI80]. They prove that a sum of these classes in a certain quotient of the Grothendieck–Witt group is a well-defined invariant of a rank n vector bundle on a suitable n-dimensional smooth projective variety [GI80, Theorem 2]. (This invariant should be viewed as an analogue of the Euler number. Recall that, on an oriented n-dimensional manifold, the Euler number of an oriented rank n vector bundle can be expressed as a sum of the local Brouwer degrees associated to a general global section.) The Main Theorem is also related to Cazanave’s work on the global A1-degree of a rational function. In [Caz08, Caz12], Cazanave proved that the global A1-degree of a 3 1 1 rational function F: Pk Pk is the class represented by the Bezout´ matrix, an explicit symmetric matrix.
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