BÉZOUTIANS AND THE A 1 -DEGREE

. We prove that both the local and global A 1 -degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that the Bézoutian bilinear form, the Scheja–Storch form, and the A 1 -degree for complete intersections are isomorphic. Our global theorem generalizes Cazanave’s theorem in the univariate case, and our local theorem generalizes Kass–Wickelgren’s theorem on EKL forms and the local degree. This result provides an algebraic formula for local and global degrees in motivic homotopy theory.


Introduction
Morel's ‫ށ‬ 1 -Brouwer degree [25] assigns a bilinear form-valued invariant to a given endomorphism of affine space.However, Morel's construction is not explicit.In order to make computations and applications, we would like algebraic formulas for the ‫ށ‬ 1 -degree.Such formulas were constructed by Cazanave for the global ‫ށ‬ 1 -degree in dimension 1 [9], Kass and Wickelgren for the local ‫ށ‬ 1 -degree at rational points and étale points [16], and Brazelton, Burklund, McKean, Montoro and Opie for the local ‫ށ‬ 1 -degree at separable points [7].In this paper, we give a general algebraic formula for the ‫ށ‬ 1 -degree in both the global and local cases.In the global case, we remove Cazanave's dimension restriction, while in the local case, we remove previous restrictions on the residue field of the point at which the local ‫ށ‬ 1 -degree is taken.
Let k be a field, and let f = ( f 1 , . . ., f n ) : ‫ށ‬ n k → ‫ށ‬ n k be an endomorphism of affine space with isolated zeros, so that Q := k[x 1 , . . ., x n ]/( f 1 , . . ., f n ) is a complete intersection.We now recall the definition of the Bézoutian of f , as well as a special bilinear form determined by the Bézoutian.Introduce new variables X := (X 1 , . . ., X n ) and Y := (Y 1 , . . ., Y n ).For each 1 ≤ i, j ≤ n, define the quantity 1986 Thomas Brazelton, Stephen McKean and Sabrina Pauli We define the Bézoutian form of f to be the class β f in the Grothendieck-Witt ring GW(k) determined by the bilinear form Q × Q → k with Gram matrix (B i, j ).
For any isolated zero of f corresponding to a maximal ideal m, there is an analogous bilinear form β f,m on the local algebra Q m .We refer to β f,m as the local Bézoutian form of f at m.We will demonstrate that both β f and β f,m yield well-defined classes in GW(k).Our main theorem is that the Bézoutian form of f agrees with the ‫ށ‬ 1 -degree in both the local and global contexts.
Theorem 1.2.Let char k ̸ = 2. Let f : ‫ށ‬ n k → ‫ށ‬ n k have an isolated zero at a closed point m.Then β f,m is isomorphic to the local ‫ށ‬ 1 -degree of f at m.If we further assume that all the zeros of f are isolated, then β f is isomorphic to the global ‫ށ‬ 1 -degree of f .
Because the Bézoutian form can be explicitly computed using commutative algebraic tools, Theorem 1.2 provides a tractable formula for ‫ށ‬ 1 -degrees and Euler classes in motivic homotopy theory.Using the Bézoutian formula for the ‫ށ‬ 1 -degree, we are able to deduce several computational rules for the degree.We also provide a Sage implementation for calculating local and global ‫ށ‬ 1 -degrees via the Bézoutian at [8].
Remark 1.3.The key contribution of this article is computability.Building on the work of Kass and Wickelgren [16], Bachmann and Wickelgren [2] show that the ‫ށ‬ 1 -degree agrees with the Scheja-Storch form as elements of KO 0 (k).In Theorem 5.1, we show how this immediately implies that the ‫ށ‬ 1 -degree and Scheja-Storch form determine the same element of GW(k).Scheja and Storch [30] showed that their form is a Bézoutian bilinear form (in the sense of Definition 3.8; see also Lemma 4.4 and Remark 4.8), which was further explored by Becker, Cardinal, Roy and Szafraniec [4].Putting these results together shows that the isomorphism class of the Bézoutian bilinear form is the ‫ށ‬ 1 -degree.
In dimension 1, Cazanave [9] gives a simple formula for computing the ‫ށ‬ 1 -degree as a Bézoutian bilinear form in the global setting.However, it is not immediately clear how to adapt this to higher dimensions or the local setting.Becker, Cardinal, Roy and Szafraniec show how to compute Bézoutian bilinear forms in terms of "dualizing forms," but this method is computationally analogous to using the Eisenbud-Khimshiashvili-Levine form to compute the ‫ށ‬ 1 -degree [16].In the proof of Theorem 1.2 (found in Section 5), we show that our two notions of Bézoutian bilinear forms (Definitions 1.1 and 3.8) agree up to isomorphism.Since Definition 1.1 is the desired generalization of Cazanave's formula, this enables us to calculate ‫ށ‬ 1 -degrees in full generality.
1A. Outline.Before proving Theorem 1.2, we recall some classical results on Bézoutians (following [4]) in Section 3, as well as the work of Scheja and Storch on residue pairings [30] in Section 4. We then discuss a local decomposition procedure for the Scheja-Storch form and show that the global Scheja-Storch form is isomorphic to the Bézoutian form in Section 4A.In Section 5, we complete the proof of Theorem 1.2 by applying the work of Kass and Wickelgren [16] and Bachmann and Wickelgren [2] on the local ‫ށ‬ 1 -degree and the Scheja-Storch form.Using Theorem 1.2, we give an algorithm for computing the local and global ‫ށ‬ 1 -degree at the end of Section 5A, available at [8].In Section 6, we establish some basic properties for computing degrees.In Section 7, we provide a step-by-step illustration of our ideas by working through some explicit examples.Finally, we implement our code to compute some examples of ‫ށ‬ 1 -Euler characteristics of Grassmannians in Section 8. We check our computations by proving a general formula for the ‫ށ‬ 1 -Euler characteristic of a Grassmannian in Theorem 8.4.The ‫ށ‬ 1 -Euler characteristic of Grassmannians is essentially a folklore result that follows from the work of Hoyois, Levine, and Bachmann and Wickelgren.
1B. Background.Let GW(k) denote the Grothendieck-Witt group of isomorphism classes of symmetric, nondegenerate bilinear forms over a field k.Morel's ‫ށ‬ 1 -Brouwer degree [25,Corollary 1.24] deg : which is a group isomorphism (in fact, a ring isomorphism [24, Lemma 6.3.8]) for n ≥ 2, demonstrates that bilinear forms play a critical role in motivic homotopy theory.However, Morel's ‫ށ‬ 1 -degree is nonconstructive.Kass and Wickelgren addressed this problem by expressing the ‫ށ‬ 1 -degree as a sum of local degrees [17,Lemma 19] and providing an explicit formula (building on the work of Eisenbud and Levine [11] and Khimshiashvili [13]) for the local ‫ށ‬ 1 -degree [16] at rational points and étale points.This explicit formula can also be used to compute the local ‫ށ‬ 1 -degree at points with separable residue field by [7].Together, these results allow one to compute the global ‫ށ‬ 1 -degree of a morphism f : with only isolated zeros by computing the local ‫ށ‬ 1 -degrees of f over its zero locus, so long as the residue field of each point in the zero locus is separable over the base field.In the local case, Theorem 1.2 gives a commutative algebraic formula for the local ‫ށ‬ 1 -degree at any closed point.
Cazanave showed that the Bézoutian gives a formula for the global ‫ށ‬ 1 -degree of any endomorphism of ‫ސ‬ 1 k [9].An advantage to Cazanave's formula is that one does not need to determine the zero locus or other local information about f .We extend Cazanave's formula for morphisms f : ‫ށ‬ n k → ‫ށ‬ n k with isolated zeros.The work of Scheja and Storch on global complete intersections [30] is central to both [16] and our result.We also rely on the work of Becker, Cardinal, Roy and Szafraniec [4], who describe a procedure for recovering the global version of the Scheja-Storch form.
Theorem 1.2 has applications wherever Morel's ‫ށ‬ 1 -degree is used.One particularly successful application of the ‫ށ‬ 1 -degree has been the ‫ށ‬ 1 -enumerative geometry program.The goal of this program is to enrich enumerative problems over arbitrary fields by producing GW(k)-valued enumerative equations and interpreting them geometrically over various fields.Notable results in this direction include Srinivasan and Wickelgren's count of lines meeting four lines in three-space [31], Larson and Vogt's count of bitangents to a smooth plane quartic [19], and Bethea, Kass, and Wickelgren's enriched Riemann-Hurwitz formula [5].See [22; 26] for other related works.For a more detailed account of recent developments in ‫ށ‬ 1 -enumerative geometry; see [6; 28].

Notation and conventions
In this section, we fix some standard terminology and notation.Let k denote an arbitrary field.We will to denote an endomorphism of affine space, assumed to have isolated zeros when we work with it in the global context.We denote by Q the global algebra associated to this endomorphism The maximal ideals of Q correspond to the maximal ideals of k[x 1 , . . ., x n ] on which f vanishes.For any maximal ideal m of k[x 1 , . . ., x n ] on which f vanishes, we denote by Q m the local algebra If λ : V → k is a k-linear form on any k-algebra, we will denote by λ the associated bilinear form given by Definition 2.1.We say that λ is a dualizing linear form if λ is nondegenerate as a symmetric bilinear form [4, 2.1].If λ is dualizing, then we say that two vector space bases {a i } and where δ i j = 1 for i = j and δ i j = 0 for i ̸ = j.We show in Remark 3.6 that if {a i } and {b i } are dual with respect to λ, then λ is a dualizing linear form.
More notation will be introduced as we provide an overview of Bézoutians and the Scheja-Storch bilinear form.We will borrow and clarify notation from both [30] and [4].

Bézoutians
We first provide an overview of the construction of the Bézoutian, following [4].Given one of our n polynomials f i , we introduce two sets of auxiliary indeterminants and study how f i changes when we incrementally exchange one set of indeterminants for the other.Explicitly, consider variables X := (X 1 , . . ., X n ) and Y := (Y 1 , . . ., Y n ).For any 1 ≤ i, j ≤ n, we denote by i j the quantity Note that i j is a multivariate polynomial.Indeed, differ only in the terms in which X j or Y j appear, so we can expand the difference where We view i j as living in the tensor product ring Q ⊗ k Q, under the isomorphism given by sending X i to x i ⊗ 1, and Y i to 1 ⊗ x i .Definition 3.1.We define the Bézoutian of the polynomials f 1 , . . ., f n to be the image Béz( f 1 , . . ., f n ) of the determinant det 3 ).Then we have that There is a natural multiplication map δ : Proof.Note that (δ • ε)(a(X, Y )) = a(x, x) and δ • ε is an algebra homomorphism.In particular, δ • ε preserves the multiplication and addition occurring in the determinant which defines Béz( f 1 , . . ., f n ).
Therefore it suffices for us to verify that Recall that Taking the x j -Taylor expansion of f (x 1 , . . ., x n ) about Y j gives us We now subtract f i (x 1 , . . ., Y j , . . ., x n ) from both sides, evaluate x j → X j , and divide by X j − Y j to deduce Finally, evaluating X j → x j and Y j → x j gives us Let a 1 , . . ., a m be any vector space basis for Q, and write the Bézoutian as Proof.This is [4, 2.10(iii)].□ This allows us to associate to the Bézoutian a pair of vector space bases for Q.Given any such pair of bases, we will construct a unique linear form for which the bases are dual.Before doing so, we establish some equivalent conditions for the duality of a linear form given a pair of bases.Proposition 3.5.Let {a i } and {b i } be a pair of bases for B. Consider the induced k-linear isomorphism Given a linear form λ : Q → k, the following are equivalent: (1) We have that (λ (2) For any a ∈ Q, we have a = i λ(aa i )b i .
(3) We have that {a i } and {b i } are dual with respect to λ.
Proof.Note that (2) implies (1) by setting a = 1.Next, we remark that is a Q-module isomorphism by [30,3.3 Satz], where the Q-module structure on Hom k (Q, k) is given by a • ϕ = ϕ(a • −).This allows us to conclude that a • (λ) = (a • λ) for any linear form λ. In particular, we have It follows from this identity that (1) implies (2).Now suppose that (2) holds.By setting a = b j for some j, we have Since {b i } is a basis, it follows that λ(a i b j ) = δ i j .Thus the bases {a i } and {b i } are dual with respect to λ.Finally, suppose that (3) holds, so that λ(a i b j ) = δ i j .For any a ∈ Q, write a as a := j c j b j for some scalars c j .Then Thus (3) implies (2).□ Remark 3.6.If {a i } and {b i } are dual with respect to λ, then λ is a dualizing form.Indeed, suppose there exists x ∈ Q such that λ (x, y) = 0 for all y ∈ Q. Write x = i x i a i with x i ∈ k.Then for all j, so x = 0.
Corollary 3.7.Let {a i } and {b i } be two k-vector space bases for Q.Then there exists a unique dualizing linear form λ : Q → k such that {a i } and {b i } are dual with respect to λ.
Proof.As is a k-algebra isomorphism, it admits a unique preimage of 1.Thus, given any pair of bases {a i } and {b i } of Q, there is a unique dualizing linear form with respect to which {a i } and {b i } are dual.□ Definition 3.8.We call λ a Bézoutian bilinear form if λ : Q → k is a dualizing linear form such that where {a i } and {b i } are dual bases with respect to λ.
A priori this is different than the Bézoutian form detailed in Definition 1.1, although we will prove that they define the same class in GW(k) in Section 5A.Proposition 3.9.Given a function f : ‫ށ‬ n k → ‫ށ‬ n k with isolated zeros, its Bézoutian bilinear form is a well-defined class in GW(k).
Proof.Let λ be a Bézoutian bilinear form for f .Recall that λ : Q × Q → k is defined by λ (a, b) = λ(ab).Since λ is a dualizing linear form, λ is nondegenerate and as Q is commutative, λ is symmetric.Lemma 3.4 implies that given a basis a 1 , . . ., a m for Q, we can write and obtain a second basis b 1 , . . ., b m for Q.By Corollary 3.7, there is a dualizing linear form for the two bases {a i } m i=1 and for some bases {a i }, {b i } dual with respect to λ and {a ′ i }, {b ′ i } dual with respect to λ ′ , then λ and λ ′ are isomorphic.We will in fact show that λ = λ ′ , so that λ = λ ′ .Write Similarly, we have λ 3 ), so that We give two bases for k 3 ) in the following table, where we replace Z by either X or Y .We pair off these bases in a convenient way.

The Scheja-Storch bilinear form
Associated to any polynomial with an isolated zero, Eisenbud and Levine [11] and Khimshiashvili [13] used the Scheja-Storch construction [30] to produce a bilinear form on the local algebra Q m .Kass and Wickelgren proved that this Eisenbud-Khimshiashvili-Levine bilinear form computes the local ‫ށ‬ 1 -degree [16].The machinery of Scheja and Storch works in great generality; in particular, one may produce a Scheja-Storch bilinear form on the global algebra Q as well as the local algebras Q m .We will provide a brief account of the Scheja-Storch construction before comparing it with the Bézoutian.In [30], k⟨X ⟩ := k⟨X 1 , . . ., X n ⟩ denotes either a polynomial ring k[X 1 , . . ., X n ] or a power series ring k[[X 1 , . . ., X n ]].We will also use this notation, although we will focus on the situation where k⟨X ⟩ is a polynomial ring.Let ρ : k⟨X ⟩ → Q denote the map obtained by quotienting out by the ideal ( f 1 , . . ., f n ), let µ 1 : k⟨X ⟩ ⊗ k k⟨X ⟩ → k⟨X ⟩ denote the multiplication map, and let µ : Q ⊗ k Q → Q denote the multiplication map on the global algebra, fitting into a commutative diagram: We remark that f j ⊗ 1 − 1 ⊗ f j lies in ker(µ 1 ), and that ker(µ 1 ) is generated by elements of the form Thus for any j, there are elements a i j ∈ k⟨X ⟩ ⊗ k k⟨X ⟩ such that We denote by the following distinguished element in the tensor algebra which corresponds to the Bézoutian which we will later demonstrate.It is true that is independent of the choice of a i j , as shown by Scheja and Storch [30, 3.1 Satz].We now define an important isomorphism χ of k-algebras used in the Scheja-Storch construction.However, we will phrase this more categorically than in [30], as it will benefit us later.
Proposition 4.1.Consider two endofunctors F, G : Alg k on the category of finitely generated k-algebras, where F(A) = A ⊗ k A and G(A) = Hom k (Hom k (A, k), A).Then there is a natural isomorphism χ : F → G whose component at a k-algebra A is Proof.This canonical isomorphism is given in [30, page 181], so it will suffice for us to verify naturality.Let g : A → B be any morphism of k-algebras.Consider the induced maps g ⊗ g : It remains to show that the following diagram commutes: To see this, we compute We now let := χ Q ( ) denote the image of under the component of this natural isomorphism at the global algebra Q.We have that is a k-linear map : , we obtain a well-defined linear form η : Q → k by [30, 3.3 The Bézoutian gives us an explicit formula for .As a result, the global Scheja-Storch form agrees with the Bézoutian form.
Proof.We showed in Proposition 4.3 that is the Bézoutian in Q ⊗ k Q.We now show that the associated forms are identical.Pick bases {a i } and Since the natural isomorphism χ has k-linear components, is mapped to Thus η := −1 (1) is the linear form η : Q → k satisfying m i=1 η(a i )b i = 1.By Proposition 3.5, this implies that η is the form for which {a i } and {b i } are dual bases.As in Definition 3.8, this tells us that η is the linear form producing the Bézoutian bilinear form.□ 4A.Local decomposition.While our discussion of the Scheja-Storch form in the previous section was global, it is perfectly valid to localize at a maximal ideal and repeat the story again [30, pages 180-181].
The fact that Q is an Artinian ring then gives a convenient way to relate the global version of η to the local version of η.This local decomposition has been utilized previously, for example in [16].
Let m be a maximal ideal in k[x 1 , . . ., In k⟨X ⟩ m ⊗ k k⟨X ⟩ m , we can again write Tracing through this diagram, we see that where m = χ Q m ( m ).Unwinding m = λ m * ( ), we find that m is the map Recall that as Q is a zero-dimensional Noetherian commutative k-algebra, the localization maps induce a k-algebra isomorphism: 1  (λ m ) m : This is reflected by an internal decomposition of Q in terms of orthogonal idempotents [4, 2.13], which we now describe; see also [32, Lemma 00JA].By the Chinese remainder theorem, we may pick a collection of pairwise orthogonal idempotents {e m } m such that m e m = 1.The internal decomposition of Q is then and the localization maps restrict to isomorphisms Since ℓ factors through the localization, it can be written as a composite Returning to the Scheja-Storch form, we have the following commutative diagram relating m and : This coherence between and m allows us to relate the local linear forms η m := −1 m (1) to the global linear form η := −1 (1) in the following way.
Proof.For each maximal ideal m, let {w m,i } i be a k-vector space basis for Q m .Let {v m,i } m,i (ranging over all i and all maximal ideals) be a basis of Q such that λ m (v m,i ) = w m,i for each i and m, and λ m (v n,i ) = 0 for m ̸ = n.We now compare the Gram matrix for η : Q → k and the Gram matrices for η m : Q m → k in these bases.Via the internal decomposition consisting of pairwise orthogonal idempotents, we have so the Gram matrix for η will be a block sum indexed over the maximal ideals.If m = n, then Proposition 4.6 implies

Proof of Theorem 1.2
We now relate the Scheja-Storch form to the ‫ށ‬ 1 -degree.The following theorem was first proven in the case where p is a rational zero by Kass and Wickelgren [16], and then in the case where p has finite separable residue field over the ground field in [7, Corollary 1.4].Recent work of Bachmann and Wickelgren [2] gives a general result about the relation between local ‫ށ‬ 1 -degrees and Scheja-Storch forms.
Theorem 5.1.Let char k ̸ = 2. Let f : ‫ށ‬ n k → ‫ށ‬ n k be an endomorphism of affine space with an isolated zero at a closed point p.Then we have that the local ‫ށ‬ 1 -degree of f at p and the Scheja-Storch form of f at p coincide as elements of GW(k): deg Proof.We may rewrite f as a section of the trivial rank n bundle over affine space O n ‫ށ‬ n k → ‫ށ‬ n k .Under the hypothesis that p is isolated, we may find a neighborhood X ⊆ ‫ށ‬ n k of p where the section f is nondegenerate (meaning it is cut out by a regular sequence).By [2, Corollary 8.2], the local index of f at p with the trivial orientation, corresponding to the representable Hermitian K -theory spectrum KO, agrees with the local Scheja-Storch form as elements of KO 0 (k): (5-1) Let ‫ޓ‬ denote the sphere spectrum in the stable motivic homotopy category SH(k).It is a well-known fact that Hermitian K -theory receives a map from the sphere spectrum, inducing an isomorphism π 0 ‫)ޓ(‬ ∼ −→ π 0 (KO) if char k ̸ = 2 (see for example [14, 6.9] for more detail); this is the only place where we use the assumption that char k ̸ = 2. Combining this with the fact that π 0 ‫)ޓ(‬ = GW(k) under Morel's degree isomorphism, we observe that (5-1) is really an equality in GW(k).By [2, Theorem 7.6, Example 7.7], the local index associated to the representable theory agrees with the local ‫ށ‬ 1 -degree: Combining these equalities gives the desired equality in GW(k).□ In contrast to previous techniques for computing the local ‫ށ‬ 1 -degree at rational or separable points, Corollary 5.3 gives an algebraic formula for the local ‫ށ‬ 1 -degree at any closed point.
As a result of the local decomposition of Scheja-Storch forms, the Bézoutian form agrees with the ‫ށ‬ 1 -degree globally as well.5A.Computing the Bézoutian bilinear form.We now prove Theorem 1.2 by describing a method for computing the class in GW(k) of the Bézoutian bilinear form in terms of the Bézoutian.
Proof of Theorem 1.2.Let R denote either a global algebra Q or a local algebra Q m .Let {α i } be any basis for R, and express (5-2) Rewriting this, we have Let β i := j B i, j α j , so that {α i } and {β i } are dual bases.Then for any linear form λ : R → k for which {α i } and {β i } are dual, we will have that λ agrees with the global or local ‫ށ‬ 1 -degree (depending on our Let λ be such a form.The product of α i and β j is given by Applying λ to each side, we get an indicator function Varying over all i, j, s, this equation above tells us that the identity matrix is equal to the product of the matrix (B j,s ) and the matrix (λ(α i α s )) = (λ(α s α i )).Explicitly, we have that .
Thus the Gram matrix for λ in the basis {α i } is (B i, j ) −1 .We conclude by proving that (B i, j ) and (B i, j ) −1 represent the same element of GW(k).Since any symmetric bilinear form can be diagonalized, T is diagonal with entries inverse to the diagonal entries of S T • (B i, j ) • S. Applying the equality ⟨a⟩ = ⟨1/a⟩ along the diagonals, it follows that (B i, j ) −1 and (B i, j ) define the same element in GW(k).Theorem 1.2 now follows from Corollaries 5.3 and 5.4.□ The following tables describe algorithms for computing the global and local ‫ށ‬ 1 -degrees in terms of the Bézoutian bilinear form.A Sage implementation of these algorithms is available at [8].
Computing the global ‫ށ‬ 1 -degree via the Bézoutian (1) Compute the i j and the image of their determinant Béz( f (2) Pick a k-vector space basis a 1 , . . ., (3) The matrix B = (B i, j ) represents deg ‫ށ‬ 1 ( f ).Diagonalize B to write its class in GW(k).
Computing the local ‫ށ‬ 1 -degree via the Bézoutian (1) Compute the i j and the image of their determinant Béz( f (2) Pick a k-vector space basis a 1 , . . ., (3) The matrix B = (B i, j ) represents deg ‫ށ‬ 1 m ( f ).Diagonalize B to write its class in GW(k).

Calculation rules
Using the Bézoutian characterization of the ‫ށ‬ 1 -degree, we are able to establish various calculation rules for local and global ‫ށ‬ 1 -degrees.See [18; 29] for related results in the local case.Our ultimate goal in this section is the product rule for the ‫ށ‬ 1 -degree (see Proposition 6.5), which was already known by the work of Morel.See the paragraph preceding Proposition 6.5 for a more detailed discussion.Proposition 6.1.Suppose that f = ( f 1 , . . ., f n ) and g = (g 1 , . . ., g n ) are endomorphisms of affine space that generate the same ideal Proof.We may choose the same basis for Q = k[x 1 , . . ., x n ]/I (or Q p in the local case) in our computation for the degrees of f and g.The Bézoutians Béz( f ) = Béz(g) will have the same coefficients in this basis, so their Gram matrices will coincide.□ The following result is the global analogue of [29,Lemma 14].
k be an endomorphism of ‫ށ‬ n k with only isolated zeros.Let A ∈ k n×n be an invertible matrix.Then Proof.Write A = (a i j ) and where g is either f or A • f .Then = n l=1 a il f l j , and thus ( ) and det A • det( f i j ) are equal.Thus the Gram matrix of the Bézoutian bilinear form for A • f is det A times the Gram matrix of the Bézoutian bilinear form for f .Proposition 6.1 then proves the claim.□ Example 6.3.We may apply Lemma 6.2 in the case where A is a permutation matrix associated to some permutation σ ∈ n .Letting f σ := ( f σ (1) , . . ., f σ (n) ), we observe that at any isolated zero p of f , and an analogous statement is true for global degrees as well.
Next, we prove a lemma inspired by [18, Lemma 12].Lemma 6.4.Let f, g : ‫ށ‬ n k → ‫ށ‬ n k be two endomorphisms of ‫ށ‬ n k .Assume that f and g are quasifinite.Let L ∈ M n (k) be an invertible n × n matrix, which defines a morphism L : ‫ށ‬ n k → ‫ށ‬ n k given by (x 1 , . . ., x n ) → (x 1 , . . ., x n ) • L T .Let I n denote the n × n identity matrix, and assume that det( Proof.Quasifinite morphisms have isolated zero loci by [32, Definition 01TD (3)].The composition of quasifinite morphisms is again quasifinite [32, Lemma 01TL], so f • g has isolated zero locus.
Next, we show that L is also quasifinite.We will actually prove a stronger statement.Let A t ∈ M n (k[t]) be an invertible n × n matrix, which implies that det Quasicompact and locally quasifinite morphisms are quasifinite [32, Lemma 01TJ], so we conclude that A t 0 is quasifinite for each t 0 ∈ ‫ށ‬ 1 k .Just as in [18,Lemma 12], we now define Then [30, page 182] gives us a Scheja-Storch form η : is symmetric and nondegenerate.By Harder's theorem [16,Lemma 30], the stable isomorphism class In particular, the Scheja-Storch bilinear forms of □ The following product rule is a consequence of Morel's proof that the ‫ށ‬ 1 -degree is a ring isomorphism [24,Lemma 6.3.8].We give a more hands-on proof of this product rule.See [18,Theorem 13] and [29,Theorem 26] for an analogous proof of the product rule for local degrees at rational points.Proposition 6.5 (product rule).Let f, g : Proof.We follow the proofs of [18,Theorem 13] and [29,Theorem 26].The general idea is to mimic the Eckmann-Hilton argument [10].Let x := (x 1 , . . ., x n ) and y := (y 1 , . . ., y n ).Define f , g : × ‫ށ‬ n by f (x, y) = ( f (x), y) and g(x, y) = (g(x), y), and note that f and g are both quasifinite because f and g are quasifinite.Since ( f • g, y) and f • g define the same ideal in k[x, y] and have the same Bézoutian, we have deg , f (y)).Using Lemma 6.4 repeatedly, we will show that deg ‫ށ‬ 1 ( f • g) = deg ‫ށ‬ 1 (g × f ).Let I n be the n × n identity matrix, and let One can check that By Theorem 1.2, (B i j ) and (B ′ i j ) are the Gram matrices for deg ‫ށ‬ 1 (g) and deg ‫ށ‬ 1 ( f ), respectively.Next, we have Béz(g x n , y 1 , . . ., y n ]/(g 1 (x), . . ., g n (x), f 1 (y), . . ., f n (y)).In this basis, we have so the Gram matrix of deg ‫ށ‬ 1 (g × f ) is the tensor product (B i j ) ⊗ (B ′ i j ).We thus we have an equality

Examples
We now give a few remarks and examples about computing the Bézoutian.
Remark 7.1.It is not always the case that the determinant det . Then the Bézoutian is given by However, the Bézoutian is symmetric once we pass to the quotient k Continuing the present example, let {1, x 2 } be a basis for the algebra which is symmetric.Moreover, the Bézoutian bilinear form is represented by 0 , where p is an odd prime, and consider the endomorphism of the affine plane given by f : As the residue field of the zero of f is not separable over k, existing strategies for computing the local ‫ށ‬ 1 -degree are insufficient.Our results allow us to compute this ‫ށ‬ 1 -degree.The Bézoutian is given by In the basis {1, x 1 , . . ., } of Q, the Bézoutian bilinear form consists of a t in the upper left corner and a 1 in each entry just below the antidiagonal.Thus 2 − a/b).We will use Bézoutians to compute the local degrees deg ‫ށ‬ 1 m ( f ) and deg ‫ށ‬ 1 n ( f ), as well as the global degree deg .
We first compute the global Bézoutian as In the basis {1, We now write the Bézoutian matrix given by the coefficients of Béz( f ): One may check (e.g., with a computer) that this is equal to ‫ވ3‬ in GW(k).
In Q m , we have that x 2 1 x 2 = x 1 x 2 = 0 and x 3 1 = b a x 1 x 2 2 = 0.In the basis {1, We thus get the Bézoutian matrix at m: This is ‫ވ‬ + ⟨a, b⟩ in GW(k).
In Q n , we have x 1 = 1.In the basis {1, x 2 } for Q n , the Bézoutian reduces to We can then write the Bézoutian matrix at n: This is ⟨−a, −b⟩ in GW(k).Note that ⟨−a, −b⟩ need not be equal to ‫.ވ‬However, this does not contradict [29, Theorem 2], since n is a nonrational point.Putting these computations together, we see that

Application:
The ‫ށ‬ 1 -Euler characteristic of Grassmannians As an application of Theorem 1.2, we compute the ‫ށ‬ 1 -Euler characteristic of various low-dimensional Grassmannians in Example 8.2 and Figure 1.These computations suggest a recursive formula for the ‫ށ‬ 1 -Euler characteristic of an arbitrary Grassmannian, which we prove in Theorem 8.4.This formula is analogous to the recursive formulas for the Euler characteristics of complex and real Grassmannians.Theorem 8.4 is probably well-known, and the proof is essentially a combination of results of Hoyois, Levine, and Bachmann-Wickelgren.
8A.The ‫ށ‬ 1 -Euler characteristic.Let X be a smooth, proper k-variety of dimension n with structure map π : X → Spec k.Let p : T X → X denote the tangent bundle of X .The ‫ށ‬ 1 -Euler characteristic χ ‫ށ‬ 1 (X ) ∈ GW(k) is a refinement of the classical Euler characteristic.In particular, if k = ‫,ޒ‬ then rank χ ‫ށ‬ 1 (X ) = χ (X ‫))ރ(‬ and sgn χ ‫ށ‬ 1 (X ) = χ (X ‫.))ޒ(‬There exist several equivalent definitions of the ‫ށ‬ 1 -Euler characteristic [20; 21; 1].For example, we may define χ ‫ށ‬ 1 (X ) to be the π -pushforward of the ‫ށ‬ 1 -Euler class of the tangent bundle [20], where z : X → T X is the zero section and CH d (X, ω X/k ) is the Chow-Witt group defined by Barge and Morel [3;12].That is, Analogous to the classical case [23], the ‫ށ‬ 1 -Euler characteristic can be computed as the sum of local ‫ށ‬ 1 -degrees at the zeros of a general section of the tangent bundle using the work of Kass and Wickelgren [2;17;20].We now describe this process.Let σ be a section of T X which only has isolated zeros.For a zero x of σ , choose Nisnevich coordinates ψ : U → ‫ށ‬ n k around x. 2 Since ψ is étale, it induces an isomorphism of tangent spaces and thus yields local coordinates around x. Shrinking U if necessary, we can trivialize T X | U ∼ = U × ‫ށ‬ n k .The chosen Nisnevich coordinates (ψ, U ) and trivialization τ : In turn, this yields a distinguished section d of Hom(det T X | U , det T X | U ), which is defined by d ψ → d τ .We say that a trivialization τ is compatible with the chosen coordinates (ψ, U ) if the image of the distinguished section d under the canonical isomorphism ρ : Hom(det Definition 21].
Given a compatible trivialization τ : T X | U ∼ = U × ‫ށ‬ n k , the section σ trivializes to σ : U → ‫ށ‬ n k .We can then define the local index ind x σ at x to be the ‫ށ‬ 1 -degree of the composite Here, the first map is the collapse map, the second map is excision, the third map is induced by the Nisnevich coordinates (ψ, U ), and the fifth map is purity; see e.g., [2,Definition 7.1].By [17,Theorem 3], the ‫ށ‬ 1 -Euler characteristic is then the sum of local indices By Theorem 1.2, we may thus compute the ‫ށ‬ 1 -Euler characteristic by computing the global Bézoutian bilinear form of an appropriate map f : is equal to the residue field of the corresponding zero.If each residue field Q m i is a separable extension of k, then the ‫ށ‬ 1 -degree of f is equal to sum of the scaled trace forms Tr Q m i /k (⟨J ( f )| m i ⟩) (see e.g., [7, Definition 1.2]), where J ( f )| m i is the determinant of the Jacobian of f evaluated at the point m i .In [27] the last named author uses the scaled trace form for several ‫ށ‬ 1 -Euler number computations.However, Theorem 1.2 yields a formula for deg ‫ށ‬ 1 ( f ) for any f with only isolated zeros and without any restriction on the residue field of each zero.Moreover, we can even compute deg ‫ށ‬ 1 ( f ) without solving for the zero locus of f .8B.The ‫ށ‬ 1 -Euler characteristic of Grassmannians.Let G := Gr k (r, n) be the Grassmannian of r -planes in k n .In order to compute χ ‫ށ‬ 1 (G), we first need to describe Nisnevich coordinates and compatible trivializations for G and T G .We then need to choose a convenient section of T G and describe the resulting endomorphism ‫ށ‬ r (n−r ) k .The tangent bundle T G → G is isomorphic to p : Hom(S, Q) → G, where S → G and Q → G are the universal sub-and quotient bundles.
We now describe Nisnevich coordinates on G and a compatible trivialization of T G , following [31].Let d = r (n − r ) be the dimension of G, and let {e 1 , . . ., e n } be the standard basis of k .
The map ψ : U → ‫ށ‬ d k given by ψ(H ({x i, j } r,n−r i, j=1 )) = ({x i, j } n−r,r i, j=1 ) yields Nisnevich coordinates (ψ, U ) centered at ψ(span{e n−r +1 , . . ., e n }) = (0, . . ., 0).For the trivialization of Then { ẽ1 , . . ., ẽn } is a basis for k n , and we denote the dual basis by { φ1 , . . ., φn }.Over U , the bundles S * and Q are trivialized by { φn−r+1 , . . ., φn } and { ẽ1 , . . ., ẽn−r }, respectively.Since Let {φ 1 , . . ., φ n } be the dual basis of the standard basis {e 1 , . . ., e n } of k n .A homogeneous degree 1 polynomial α ∈ k[φ 1 , . . ., φ n ] gives rise to a section s of S * , defined by evaluating α.In particular, given a vector t = n i=1 t i ẽi in H ({x i, j } r,n−r i, j=1 ), we use the dual change of basis Taking n sections s 1 , . . ., s n of S * , we get a section of T G ∼ = Hom(S, Q) given by where the second map is quotienting by { ẽn−r+1 , . . ., ẽn }.We obtain our map ‫ށ‬ d k → ‫ށ‬ d k by applying the trivializations { φn−r+i ⊗ ẽ j } r,n−r i, j=1 of T G .Explicitly, take n sections s 1 , . . ., s n of S * .Since e i = ẽi − n−r j=1 x i−(n−r ), j e j for i > n − r , we have s j e j ≡ s j e j − r i=1 x i, j s n−r +i e j mod ( ẽn−r+1 , . . ., ẽn ), for all j ≤ n − r .Recall that e j = ẽ j for j ≤ n − r .The coordinate of ‫ށ‬ d k → ‫ށ‬ d k corresponding to φn−r+i ⊗ ẽ j is thus the coefficient of t n−r +i in s j (t) − r ℓ=1 x ℓ, j s n−r +ℓ (t).For a general section σ of p : T G → G, the finitely many zeros of σ will all lie in U .In this case, the ‫ށ‬ 1 -Euler characteristic of G is equal to the global ‫ށ‬ 1 -degree of the resulting map ‫ށ‬ d k → ‫ށ‬ d k , which can computed using the Bézoutian.
Using a computer, we performed computations analogous to Example 8.2 for r ≤ 5 and n ≤ 7.These ‫ށ‬ 1 -Euler characteristics of Grassmannians are recorded in Figure 1.
Recall that the Euler characteristics of real and complex Grassmannians are given by binomial coefficients.In particular, these Euler characteristics satisfy certain recurrence relations related to Pascal's rule.The computations in Figure 1 indicate that an analogous recurrence relation is true for the ‫ށ‬ 1 -Euler characteristic of Grassmannians over an arbitrary field.In fact, this recurrence relation is a direct consequence of a result of Levine [20].We can now apply a theorem of Bachmann and Wickelgren [2] to completely characterize χ ‫ށ‬ 1 (Gr k (r, n)).
Theorem 8.4.Let k be field of characteristic not equal to 2. Let n ‫ރ‬ := n r , and let n ‫ޒ‬ := ⌊n/2⌋ ⌊r/2⌋ .Then Proof.By [2, Theorem 5.8], we can restrict this computation to two different possibilities.We will prove by induction that χ ‫ށ‬ 1 (Gr k (r, n)) mod ‫ވ‬ has no ⟨2⟩ summand.The desired result will then follow from [2, Theorem 5.8] by noting that n ‫ރ‬ and n ‫ޒ‬ are the Euler characteristics of Gr ‫ރ‬ (r, n) and Gr ‫ޒ‬ (r, n), respectively.Since ‫ށ‬ n k is ‫ށ‬ 1 -homotopic to Spec k, we have χ ‫ށ‬  We rewrite the data recorded in Figure 1 in a modified Pascal's triangle in Figure 3.The rows correspond to the dimension n of the ambient affine space k n , while the southwest-to-northeast diagonals correspond to the dimension r of the planes k r in the ambient space.

Proposition 4 . 5 .
Suppose ℓ : Q → k is a linear form which factors through the localization λ m : Q → Q m for some maximal ideal m.Then (ℓ) lies in Q • e m .Proof.Recall that λ m | Q•e n = 0 for n ̸ = m.Since e m •e n = 0 for n ̸ = m and e m is idempotent, the localization λ m : Q → Q m can be written as the following composition:

Remark 4 . 8 .
Thus the Gram matrices of η and m η m are equal, so η = m η m .□The local Scheja-Storch bilinear form is given by ηm : Q m × Q m → k.Given a basis {a 1 , . . ., a m } of Q m ,we may write m = a i ⊗ b i and define the local Bézoutian bilinear form as a suitable dualizing form.Replacing Q, , , and η with Q m , m , m , and η m , the results of Sections 3 and 4 also hold for local Bézoutians and the local Scheja-Storch form.In particular, the local analog of Lemma 4.4 implies that the local Scheja-Storch form is equal to the local Bézoutian form.

Remark 5 . 2 . 7 Corollary 5 . 3 .
Bachmann and Wickelgren in fact show that deg ‫ށ‬ 1 Z ( f ) = SS Z ( f ) for any isolated zero locus Z of f [2, Corollary 8.2].This gives an alternate viewpoint on the local decomposition described in Lemma 4.Let char k ̸ = 2.The local Bézoutian bilinear form is the local ‫ށ‬ 1 -degree.Proof.As discussed in Remark 4.8, we can modify Lemma 4.4 to the local case by replacing Q, , , and η with Q m , m , m , and η m .The local Bézoutian form is thus equal to the local Scheja-Storch form, which is equal to the local ‫ށ‬ 1 -degree by Theorem 5.1.□

Corollary 5 . 4 .
Let char k ̸ = 2.The Bézoutian bilinear form is the global ‫ށ‬ 1 -degree.Proof.Let η denote the Bézoutian bilinear form, which is equal to the global Scheja-Storch bilinear form by Lemma 4.4.By Lemma 4.7, the global Scheja-Storch form decomposes as a block sum of local Scheja-Storch forms.By Theorem 5.1, the local Scheja-Storch bilinear form agrees with the local ‫ށ‬ 1 -degree.Finally, we have that the sum of local ‫ށ‬ 1 -degrees is the global ‫ށ‬ 1 -degree.Putting this all together, we have

2000
Thomas Brazelton, Stephen McKean and Sabrina Pauli choice of R) by Corollaries 5.3 and 5.4.

2002
Thomas Brazelton, Stephen McKean and Sabrina Pauli

2004
Thomas Brazelton, Stephen McKean and Sabrina Pauli

2006
Thomas Brazelton, Stephen McKean and Sabrina Pauli be the open affine subset consisting of the r -planes H ({x i, j } r,n−r i, j=1 ) := span e n−r +i + we get a trivialization of T G | U given by { φn−r+i ⊗ ẽ j } r,n−r i, j=1 .By construction, our Nisnevich coordinates (ψ, U ) induce this local trivialization of T G .It follows that the distinguished element of Hom(det T G | U , det T G | U ) sending the distinguished element of det T G | U (determined by the Nisnevich coordinates) to the distinguished element of T G | U (determined by our local trivialization) is just the identity, which is a square.Next, we describe sections of T G → G and the resulting endomorphisms ‫ށ‬ d k → ‫ށ‬ d k . n