Newton Transformations and the Motivic Milnor Fiber of a Plane Curve

Abstract

In this article we give an expression of the motivic Milnor fiber at infinity and the motivic nearby cycles at infinity of a polynomial f in two variables with coefficients in an algebraic closed field of characteristic zero. This expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm at infinity of f without any condition of convenience or non degeneracy. In the complex setting, we compute the Euler characteristic of the generic fiber of f in terms of the area of the surfaces associated to faces of the Newton polygons. Furthermore, if f has isolated singularities, we compute similarly the classical invariants at infinity λc(f ) which measures the non equisingularity at infinity of the fibers of f in P 2 , and we prove the equality between the topological and the motivic bifurcation sets and give an algorithm to compute them.

Newton transformations and motivic invariants at infinity of plane curves

“…We extend these definition to c = ∞, considering the rational function Q/P and defining E ∞ as the fan E 0 for Q/P . We recall the Newton algorithm and refer to [40,1,6,7,4] for more details. Definition 1.6.…”

“…More generally (and similarly to [37,1,6,4,5]), we will need to consider motivic zeta functions with differential forms. Let ν ∈ N ≥1 and a differential form ω on X, such that the restriction to the open set X is equal to x ν−1 dx ∧ dy and ν = 1 if "x = 0" is not included in I.…”

“…The rationality result and the independence of the motive S f ,ω,X ((0,0),c) from the differential form ω (then its equality with S f,(0,0),c ) are standard and follow from [11,19] (see for instance [5, §2.5]). Using the Newton algorithm and the strategy of the first author and Veys in [6] (see also [4], [17] [1]), the motivic Milnor fiber S f,(0,0),c is given in terms of Newton polygons of P − cQ and Q and their Newton transforms. Classifying the arcs (x(t), y(t)) occurring in the definition of the motivic zeta function by the condition (ord(x(t)) , ord(y(t))) ∈ C with C a cone of the fan E c , we decompose (section 5.1, formula 5.1) the motivic zeta function as…”

“…In [31], the second author proved that the set of values c such that S f,x,c ̸ = 0 is a finite set, denoted by B mot f,x and called motivic bifurcation set (Definition 4.7). In this article, similarly to [4,5], assuming k algebraically closed, we investigate the case d = 2 in full generality (namely without any assumptions of convenience or non degeneracy w.r.t any Newton polygon) using ideas of Guibert in [17], Guibert-Loeser-Merle in [18,20], and the works of the first author and Veys in the case of an ideal of k [[x, y]] in [6,7].…”

“…Definition 1.9. The Newton algorithm of a polynomial P ∈ k[x, y] is defined by induction (see for instance [4,Theorem 1,Lemma 4]). It starts by applying Newton transformations given by the equations of the faces of the Newton polygon N (P ) and the roots of the corresponding face polynomials.…”