Puiseux Pairs Research Articles

Conjectures on Stably Newton Degenerate Singularities

We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, x^p+x^q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

Yano's Conjecture for Two-Puiseux-Pair Irreducible Plane Curve Singularities

In 1982, Tamaki Yano proposed a conjecture predicting the b -exponents of an irreducible plane curve singularity germ that is generic in its equisingularity class. In this article, we prove the conjecture for the case in which the irreducible germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. This hypothesis on the monodromy implies that the b -exponents coincide with the opposite of the roots of the Bernstein polynomial, and we compute the roots of the Bernstein polynomial.

Bernstein polynomial of $2$-Puiseux pairs irreducible plane curve singularities

In 1982, Tamaki Yano proposed a conjecture predicting the set of b-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In \cite{ACLM-Yano2} we proved the conjecture for the case in which the germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. In this article we aim to study the Bernstein polynomial for any function with the hypotheses above. In particular the set of all common roots of those Bernstein polynomials is given. We provide also bounds for some analytic invariants of singularities and illustrate the computations in suitable examples.

Equivalence relations for two variable real analytic function germs

For two variable real analytic function germs we compare the blow-analytic equivalence in the sense of Kuo to the other natural equivalence relations. Our main theorem states that $C^1$ equivalent germs are blow-analytically equivalent. This gives a negative answer to a conjecture of Kuo. In the proof we show that the Puiseux pairs of real Newton-Puiseux roots are preserved by the $C^1$ equivalence of function germs. The proof is achieved, being based on a combinatorial characterisation of blow-analytic equivalence in terms of the real tree model. We also give several examples of bi-Lipschitz equivalent germs that are not blow-analytically equivalent.

Jet schemes and minimal embedded desingularization of plane branches

For a plane branch C with g Puiseux pairs, we determine the irreducible components of its jet schemes which correspond to the star (or rupture) and end divisors that appear on the dual graph of the minimal embedded desingularization of C. We exploit these informations to construct a Teissier type resolution of C embedded in $${\mathbb{C}^{g+1}}$$ , which is special in the sense that its restriction to the strict transform of the plane induces the minimal embedded desingularization of C.

Blow-analytic equivalence of two variable real analytic function germs

Blow-analytic equivalence is a notion for real analytic function germs, introduced by Tzee-Char Kuo in order to develop real analytic equisingularity theory. In this paper we give complete characterisations of blow-analytic equivalence in the two dimensional case, in terms of the real tree model for the arrangement of real parts of Newton-Puiseux roots and their Puiseux pairs, and in terms of minimal resolutions. These characterisations show that in the two dimensional case the blow-analytic equivalence is a natural analogue of topological equivalence of complex analytic function germs. Moreover, we show that in the two-dimensional case the blow-analytic equivalence can be made cascade, and hence satisfies several geometric properties. It preserves, for instance, the contact order of real analytic arcs. In the general n n -dimensional case, we show that a singular real modification satisfies the arc-lifting property.

A criterion for topological equivalence of two variable complex analytic function germs

We show that two analytic function germs $(\mathbf{C}^{2},0) \to (\mathbf{C},0)$ are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pair of distinct components.

The zeta function of a quasi-ordinary singularity

We prove that the zeta function of an irreducible hypersurface quasi-ordinary singularity f equals the zeta function of a plane curve singularity g. If the local coordinates . Moreover, the Puiseux pairs of g can also be recovered from (any set of) distinguished tuples of f.

An equivalence for irreducible parametrization and its applications to the direct proof of an equivalence of the Puiseux pairs and the multiplicity sequences for irreducible curves

An equivalence for irreducible parametrization and its applications to the direct proof of an equivalence of the Puiseux pairs and the multiplicity sequences for irreducible curves

Normal forms and bases of local rings of irreducible germs of functions of two variables

Irreducible germs of functions of two variables are studied; their normal forms relative to the action of the group of germs of diffeomorphisms are computed along with the bases of their local rings. The answers are formulated in terms of Puiseux pairs of the germ.

The signature of smoothings of complex surface singularities

Let f : (1~3, 0)-"~(t~, 0) be the germ of a complex analytic function with an isolated critical point at the origin. For e > 0 suitably small and 6 yet smaller, the space V ' = f l ( 6 ) ~ D , (where D~ denotes the closed disk of radius e about 0) is a real oriented four-manifold with boundary whose diffeomorphism type depends only on f It has been proved that V' has the homotopy type of a wedge of two-spheres; the number p o f two-spheres is readily computable. Recently an interesting formula for g was given in terms of analytic invariants of a resolution of the singularity at 0 of the complex surface f-1(0) [13]. This formula is proved by applying the Riemann-Roch theorem to the projective completions o f f l ( 0 ) and fi (6) , then canceling terms coming from the parts away from the origin. The purpose of this paper is to find a similar formula for the signature of the intersection pairing on the two-dimensional homology of the manifold V', using the Hirzebruch signature theorem instead of the Riemann-Roch theorem. Various other signature formulas are known, in higher dimensions as well as in dimension two. For f (x , y, z) of the form g(x, y) + z 2, the intersection pairing of V' is the same as a symmetrized Seifert matrix of the compound torus link {g(x, y )=0}~S 3. There is a simple formula for the signature of the symmetrized Seifert matrix of a compound link of one component [20]; hence if g-1(0) is irreducible, there is a simple formula for the signature of V' in terms of the Puiseux pairs ofg. If g1(0) has several branches at the origin, it is possible to find a Seifert matrix for the link defined by g and compute the signature [17], but this process is tedious for all but the simplest links. Formulas also exist for the signature when f(x, y, z) is of the type x a + yb + z c [10], or when f is weighted homogeneous [22]. There is in addition a formula for the signature in terms of mixed Hodge structure [233. The genus (or geometric genus) of the singularity f 1(0) (assumed Stein) is the dimension of HI(V,, C~), where ~" is a resolution of f-1(0). By combining the formulas for the signature and the number # it is possible to show that twice the genus of the singularity f 1(0) is equal to the number of positive plus the number of