The bifurcation set of a rational function via Newton polytopes

Nguyen, Tat Thang; Saito, Takahiro; Takeuchi, Kiyoshi

doi:10.1007/s00209-021-02698-7

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2021

2022

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

(1 citation statement)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They proved that the set of atypical values is finite and called it, the topological bifurcation set B top f,x of the germ of f at x. Later, several authors (for instance [33,34], [2,3], [15], [39], [26], [27]) studied singularities of meromorphic functions, locally or at infinity. This is related to the study of pencils aP + bQ = 0 (for instance [36], [29], [33], [10]).…”

Section: Introductionmentioning

confidence: 99%

Newton transformations and motivic invariants at infinity of plane curves

Cassou-Noguès

Raibaut

2021

Math. Z.

View full text Add to dashboard Cite

Let P and Q be two polynomials in two variables with coefficients in an algebraic closed field of characteristic zero. We consider the rational function f = P/Q. For an indeterminacy point x of f and a value c, we compute the motivic Milnor fiber S f,x,c in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithms of P − cQ and Q at x, without any condition of non-degeneracy or convenience. In the complex setting, assuming for any (a, b) ∈ C 2 that x is smooth or an isolated critical point of aP + bQ, and the curves P = 0 and Q = 0 do not have common irreducible component, we prove that the topological bifurcation set B top f,x is equal to the motivic bifurcation set B mot f,x and they are computed from the Newton algorithm.

show abstract