Marcelo Hernandes | Universidade Estadual de Maringa (original) (raw)

Papers by Marcelo Hernandes

arXiv (Cornell University), May 26, 2021

The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an anal... more The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an analytical invariant of a plane branch. Giving a plane branch C with semigroup Γ there are a finitely number of distinct possible sets Λ i according to the analytic class of C. In this work we show that the value set of 1-forms Λ determines the semigroup Γ and we present an effective method to recover Γ by Λ. In particular, this allows us to decide if a subset of N is a value set of 1-forms for a plane branch.

À Deus, pelo dom da vida e pela oportunidade de fazer o que gosto. Ao Prof. Dr. Abramo Hefez pela... more À Deus, pelo dom da vida e pela oportunidade de fazer o que gosto. Ao Prof. Dr. Abramo Hefez pela orientação e pela amizade consolidada nestes últimos anos. Aos meus pais, Cláudio e Izabel, minha irmã Sheila e a pequena Gabriela, que percorreram comigo mais esta caminhada. A Maria Elenice Rodrigues por todo o seu carinho e apoio. Aos velhos e novos amigos, pelo companherismo. Ao DMA-UEM, ICMC-USP e pós-graduação em Matemática da UFF, por propiciarem as condições ideais para a execução deste trabalho. A CAPES, pelo financiamento parcial deste trabalho.

arXiv (Cornell University), Mar 9, 2017

We describe an algorithm to compute a presentation of the pushforward module f * OX for a finite ... more We describe an algorithm to compute a presentation of the pushforward module f * OX for a finite map germ f : X → (C n+1 , 0), where X is Cohen-Macaulay of dimension n. The algorithm is an improvement of a method by Mond and Pellikaan. We give applications to problems in singularity theory, computed by means of an implementation in the software Singular.

Agradecimentos À Deus, pelo dom da vida. À prof. Dr. Ires Dias, que além de orientadora e profess... more Agradecimentos À Deus, pelo dom da vida. À prof. Dr. Ires Dias, que além de orientadora e professora, se tornou uma grande amiga. À minha noiva Joseane, que compreendeu minha ausência e me incentivou neste período através de seu carinho. Aos meus pais Cláudio e Izabel, minha irmã Sheila, que mesmo à distância demonstraram seu apoio. Ao prof. Dr. Antônio Conde, pelas sugestões e atenção dispensada nas discussões de alguns fatos. Ao prof. Dr. Antônio Pactues e ao prof. Dr. Clotilzio Moreira dos Santos pelas inúmeras sugestões, que contribuiram para esta versão final. À CAPES, pelo financiamento parcial deste trabalho. À Universidade Estadual de Maringá, pela liberação parcial de minhas atividades docentes. Aos velhos e novos amigos, pelo apoio e companherismo. A todos meus professores, pelos conhecimentos transmitidos. Aos funcionários do ICMSC e UEM, em todos os escalões, que direta ou indiretamente contribuiram para o andamento e conclusão deste trabalho. Resumo Neste trabalho apresentamos a descrição dos grupos de Witt das formas A-hermitianas sobre álgebras de quaternios sobre corpos Hilbertianos, corpos finitos e extensões destes, corpos de números p-ádicos e o corpo dos números racionais.

arXiv (Cornell University), Apr 3, 2023

In this work we describe dicritical foliations in (C 2 , 0) at a triple point of the resolution d... more In this work we describe dicritical foliations in (C 2 , 0) at a triple point of the resolution dual graph of an analytic plane branch C F using its semiroots. In particular, we obtain a constructive method to present a one-parameter family C Fu of separatrices for such foliations. As a by-product we relate the contact order between a special member of C Fu and C F with analytic discrete invariants of plane branches.

Czechoslovak Mathematical Journal, Nov 10, 2009

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

arXiv (Cornell University), Apr 22, 2021

The value semigroup of a k-semiroot C k of a plane branch C allows us to recover part of the valu... more The value semigroup of a k-semiroot C k of a plane branch C allows us to recover part of the value semigroup Γ = v 0 ,. .. , v g of C, that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λ k of C k , that is an analytical invariant, and we show how it determine part of the set of values of differentials Λ of C. As a consequence, in a fixed topological class, we relate the Tjurina number τ of C with the Tjurina number of C k. In particular, we show that τ ≤ µ − 3ng −2 4 µ g−1 where n g = gcd(v 0 ,. .. , v g−1), µ and µ g−1 denote the Milnor number of C and C g−1 respectively. If n g = 2, we have that τ = µ − µ g−1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister.

arXiv (Cornell University), Oct 10, 2020

In this paper, we present a solution to the problem of the analytic classification of germs of pl... more In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allow us to recover some particular cases found in the literature.

arXiv (Cornell University), Apr 17, 2017

We introduce the semiring of values Γ with respect to the tropical operations associated to an al... more We introduce the semiring of values Γ with respect to the tropical operations associated to an algebroid curve. As a set, Γ determines and is determined by the well known semigroup of values S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve.

arXiv (Cornell University), Mar 11, 2019

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the... more In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n-space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I = O, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module Ω O/C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of Ω O/C , which is an important analytic invariant associated to the curve.

Semigroup Forum, Jul 11, 2022

The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an anal... more The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an analytical invariant of a plane branch. Giving a plane branch C with semigroup Γ there are a finitely number of distinct possible sets Λ i according to the analytic class of C. In this work we show that the value set of 1-forms Λ determines the semigroup Γ and we present an effective method to recover Γ by Λ. In particular, this allows us to decide if a subset of N is a value set of 1-forms for a plane branch.

Mathematical proceedings of the Cambridge Philosophical Society, Nov 9, 2016

The aim of this work is to characterise the k-polar variety Pk(X) of an (n − 1)-ruled hypersurfac... more The aim of this work is to characterise the k-polar variety Pk(X) of an (n − 1)-ruled hypersurface X ⊂ ℂ n+1. More precisely, we prove that Pk(X) is empty for all k >1 and the first polar variety is empty or it is an (n − 2)-ruled variety in ℂ n+1, whose multiplicity is obtained by the multiplicity of the base curve and the multiplicity of one directrix of X. As a consequence we obtain the Euler obstruction Eu 0(X) of X and, in addition, we exhibit (n − 1)-ruled hypersurfaces such that Eu 0(X) = m, for any prescribed positive integer m.

International Journal of Mathematics

In this paper, we introduce a distinguished expression for a given 1-form with respect to a polyn... more In this paper, we introduce a distinguished expression for a given 1-form with respect to a polynomial [Formula: see text], called Weierstrass form. We will use this form and the properties of plane analytical curves to give new characterizations of nondicritical generalized curve foliations.

arXiv (Cornell University), Apr 17, 2017

We introduce the semiring of values Γ with respect to the tropical operations associated to an al... more We introduce the semiring of values Γ with respect to the tropical operations associated to an algebroid curve. As a set, Γ determines and is determined by the well known semigroup of values S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve.

Journal of Algebra, 2022

The value semigroup of a k-semiroot C k of a plane branch C allows us to recover part of the valu... more The value semigroup of a k-semiroot C k of a plane branch C allows us to recover part of the value semigroup Γ = v 0 ,. .. , v g of C, that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λ k of C k , that is an analytical invariant, and we show how it determine part of the set of values of differentials Λ of C. As a consequence, in a fixed topological class, we relate the Tjurina number τ of C with the Tjurina number of C k. In particular, we show that τ ≤ µ − 3ng −2 4 µ g−1 where n g = gcd(v 0 ,. .. , v g−1), µ and µ g−1 denote the Milnor number of C and C g−1 respectively. If n g = 2, we have that τ = µ − µ g−1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister.

Journal of Algebra, 2020

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the... more In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n-space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I = O, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module Ω O/C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of Ω O/C , which is an important analytic invariant associated to the curve.

International Journal of Mathematics, 2018

To an equisingularity class of complex plane branches, described by its multiplicity [Formula: se... more To an equisingularity class of complex plane branches, described by its multiplicity [Formula: see text] and characteristic exponents [Formula: see text], [Formula: see text], there is a naturally associated family [Formula: see text] of equations containing a complete set of analytic representatives for all branches of the class. We show in this paper that the general polar curve of any member of [Formula: see text] is Newton degenerate, except when [Formula: see text], in which case the general member of [Formula: see text] corresponds to a curve which has a Newton non-degenerate general polar curve with a fixed Newton polygon, or when [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text] and [Formula: see text] is odd, in which case [Formula: see text] has a subset containing a complete set of analytic representatives for all branches of the class whose general member has also a Newton non-degenerate general polar curve with ...

Singularities and Foliations. Geometry, Topology and Applications, 2018

In this paper we present the most complete description as possible of the factorization of the ge... more In this paper we present the most complete description as possible of the factorization of the general polar of the general member of an equisingularity class of irreducible germs of complex plane curves. Our result will refine the rough description of the factorization given by M. Merle in [M] and it is based on the result given by E. Casas-Alvero in [C2] that describes the cluster of the singularities of such polars. By using our analysis, it will be possible to characterize all equisingularity classes of irreducible plane germs with r characteristic exponents having the exceptional behavior that the general polar of a general curve in this equisingularity class has only irreducible components with less than r characteristic exponents, generalizing a result obtained for r = 2 in [HHI1].

arXiv (Cornell University), Apr 3, 2023

Transactions of the American Mathematical Society, 2019

We introduce the concept of good Saito's basis for a plane curve S and we explore it to obtain a ... more We introduce the concept of good Saito's basis for a plane curve S and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we present a positive answer for a question of Dimca and Greuel relating the Tjurina number and the Milnor number for a singular irreducible plane curve.

arXiv (Cornell University), May 26, 2021

The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an anal... more The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an analytical invariant of a plane branch. Giving a plane branch C with semigroup Γ there are a finitely number of distinct possible sets Λ i according to the analytic class of C. In this work we show that the value set of 1-forms Λ determines the semigroup Γ and we present an effective method to recover Γ by Λ. In particular, this allows us to decide if a subset of N is a value set of 1-forms for a plane branch.

À Deus, pelo dom da vida e pela oportunidade de fazer o que gosto. Ao Prof. Dr. Abramo Hefez pela... more À Deus, pelo dom da vida e pela oportunidade de fazer o que gosto. Ao Prof. Dr. Abramo Hefez pela orientação e pela amizade consolidada nestes últimos anos. Aos meus pais, Cláudio e Izabel, minha irmã Sheila e a pequena Gabriela, que percorreram comigo mais esta caminhada. A Maria Elenice Rodrigues por todo o seu carinho e apoio. Aos velhos e novos amigos, pelo companherismo. Ao DMA-UEM, ICMC-USP e pós-graduação em Matemática da UFF, por propiciarem as condições ideais para a execução deste trabalho. A CAPES, pelo financiamento parcial deste trabalho.

arXiv (Cornell University), Mar 9, 2017

We describe an algorithm to compute a presentation of the pushforward module f * OX for a finite ... more We describe an algorithm to compute a presentation of the pushforward module f * OX for a finite map germ f : X → (C n+1 , 0), where X is Cohen-Macaulay of dimension n. The algorithm is an improvement of a method by Mond and Pellikaan. We give applications to problems in singularity theory, computed by means of an implementation in the software Singular.

Agradecimentos À Deus, pelo dom da vida. À prof. Dr. Ires Dias, que além de orientadora e profess... more Agradecimentos À Deus, pelo dom da vida. À prof. Dr. Ires Dias, que além de orientadora e professora, se tornou uma grande amiga. À minha noiva Joseane, que compreendeu minha ausência e me incentivou neste período através de seu carinho. Aos meus pais Cláudio e Izabel, minha irmã Sheila, que mesmo à distância demonstraram seu apoio. Ao prof. Dr. Antônio Conde, pelas sugestões e atenção dispensada nas discussões de alguns fatos. Ao prof. Dr. Antônio Pactues e ao prof. Dr. Clotilzio Moreira dos Santos pelas inúmeras sugestões, que contribuiram para esta versão final. À CAPES, pelo financiamento parcial deste trabalho. À Universidade Estadual de Maringá, pela liberação parcial de minhas atividades docentes. Aos velhos e novos amigos, pelo apoio e companherismo. A todos meus professores, pelos conhecimentos transmitidos. Aos funcionários do ICMSC e UEM, em todos os escalões, que direta ou indiretamente contribuiram para o andamento e conclusão deste trabalho. Resumo Neste trabalho apresentamos a descrição dos grupos de Witt das formas A-hermitianas sobre álgebras de quaternios sobre corpos Hilbertianos, corpos finitos e extensões destes, corpos de números p-ádicos e o corpo dos números racionais.

arXiv (Cornell University), Apr 3, 2023

In this work we describe dicritical foliations in (C 2 , 0) at a triple point of the resolution d... more In this work we describe dicritical foliations in (C 2 , 0) at a triple point of the resolution dual graph of an analytic plane branch C F using its semiroots. In particular, we obtain a constructive method to present a one-parameter family C Fu of separatrices for such foliations. As a by-product we relate the contact order between a special member of C Fu and C F with analytic discrete invariants of plane branches.

Czechoslovak Mathematical Journal, Nov 10, 2009

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

arXiv (Cornell University), Apr 22, 2021

The value semigroup of a k-semiroot C k of a plane branch C allows us to recover part of the valu... more The value semigroup of a k-semiroot C k of a plane branch C allows us to recover part of the value semigroup Γ = v 0 ,. .. , v g of C, that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λ k of C k , that is an analytical invariant, and we show how it determine part of the set of values of differentials Λ of C. As a consequence, in a fixed topological class, we relate the Tjurina number τ of C with the Tjurina number of C k. In particular, we show that τ ≤ µ − 3ng −2 4 µ g−1 where n g = gcd(v 0 ,. .. , v g−1), µ and µ g−1 denote the Milnor number of C and C g−1 respectively. If n g = 2, we have that τ = µ − µ g−1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister.

arXiv (Cornell University), Oct 10, 2020

In this paper, we present a solution to the problem of the analytic classification of germs of pl... more In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allow us to recover some particular cases found in the literature.

arXiv (Cornell University), Apr 17, 2017

We introduce the semiring of values Γ with respect to the tropical operations associated to an al... more We introduce the semiring of values Γ with respect to the tropical operations associated to an algebroid curve. As a set, Γ determines and is determined by the well known semigroup of values S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve.

arXiv (Cornell University), Mar 11, 2019

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the... more In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n-space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I = O, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module Ω O/C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of Ω O/C , which is an important analytic invariant associated to the curve.

Semigroup Forum, Jul 11, 2022

The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an anal... more The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an analytical invariant of a plane branch. Giving a plane branch C with semigroup Γ there are a finitely number of distinct possible sets Λ i according to the analytic class of C. In this work we show that the value set of 1-forms Λ determines the semigroup Γ and we present an effective method to recover Γ by Λ. In particular, this allows us to decide if a subset of N is a value set of 1-forms for a plane branch.

Mathematical proceedings of the Cambridge Philosophical Society, Nov 9, 2016

The aim of this work is to characterise the k-polar variety Pk(X) of an (n − 1)-ruled hypersurfac... more The aim of this work is to characterise the k-polar variety Pk(X) of an (n − 1)-ruled hypersurface X ⊂ ℂ n+1. More precisely, we prove that Pk(X) is empty for all k >1 and the first polar variety is empty or it is an (n − 2)-ruled variety in ℂ n+1, whose multiplicity is obtained by the multiplicity of the base curve and the multiplicity of one directrix of X. As a consequence we obtain the Euler obstruction Eu 0(X) of X and, in addition, we exhibit (n − 1)-ruled hypersurfaces such that Eu 0(X) = m, for any prescribed positive integer m.

International Journal of Mathematics

In this paper, we introduce a distinguished expression for a given 1-form with respect to a polyn... more In this paper, we introduce a distinguished expression for a given 1-form with respect to a polynomial [Formula: see text], called Weierstrass form. We will use this form and the properties of plane analytical curves to give new characterizations of nondicritical generalized curve foliations.

arXiv (Cornell University), Apr 17, 2017

We introduce the semiring of values Γ with respect to the tropical operations associated to an al... more We introduce the semiring of values Γ with respect to the tropical operations associated to an algebroid curve. As a set, Γ determines and is determined by the well known semigroup of values S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve.

Journal of Algebra, 2022

The value semigroup of a k-semiroot C k of a plane branch C allows us to recover part of the valu... more The value semigroup of a k-semiroot C k of a plane branch C allows us to recover part of the value semigroup Γ = v 0 ,. .. , v g of C, that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λ k of C k , that is an analytical invariant, and we show how it determine part of the set of values of differentials Λ of C. As a consequence, in a fixed topological class, we relate the Tjurina number τ of C with the Tjurina number of C k. In particular, we show that τ ≤ µ − 3ng −2 4 µ g−1 where n g = gcd(v 0 ,. .. , v g−1), µ and µ g−1 denote the Milnor number of C and C g−1 respectively. If n g = 2, we have that τ = µ − µ g−1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister.

Journal of Algebra, 2020

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the... more In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n-space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I = O, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module Ω O/C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of Ω O/C , which is an important analytic invariant associated to the curve.

International Journal of Mathematics, 2018

To an equisingularity class of complex plane branches, described by its multiplicity [Formula: se... more To an equisingularity class of complex plane branches, described by its multiplicity [Formula: see text] and characteristic exponents [Formula: see text], [Formula: see text], there is a naturally associated family [Formula: see text] of equations containing a complete set of analytic representatives for all branches of the class. We show in this paper that the general polar curve of any member of [Formula: see text] is Newton degenerate, except when [Formula: see text], in which case the general member of [Formula: see text] corresponds to a curve which has a Newton non-degenerate general polar curve with a fixed Newton polygon, or when [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text] and [Formula: see text] is odd, in which case [Formula: see text] has a subset containing a complete set of analytic representatives for all branches of the class whose general member has also a Newton non-degenerate general polar curve with ...

Singularities and Foliations. Geometry, Topology and Applications, 2018

In this paper we present the most complete description as possible of the factorization of the ge... more In this paper we present the most complete description as possible of the factorization of the general polar of the general member of an equisingularity class of irreducible germs of complex plane curves. Our result will refine the rough description of the factorization given by M. Merle in [M] and it is based on the result given by E. Casas-Alvero in [C2] that describes the cluster of the singularities of such polars. By using our analysis, it will be possible to characterize all equisingularity classes of irreducible plane germs with r characteristic exponents having the exceptional behavior that the general polar of a general curve in this equisingularity class has only irreducible components with less than r characteristic exponents, generalizing a result obtained for r = 2 in [HHI1].

arXiv (Cornell University), Apr 3, 2023

Transactions of the American Mathematical Society, 2019

We introduce the concept of good Saito's basis for a plane curve S and we explore it to obtain a ... more We introduce the concept of good Saito's basis for a plane curve S and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we present a positive answer for a question of Dimca and Greuel relating the Tjurina number and the Milnor number for a singular irreducible plane curve.