Herwig Hauser
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Herwig Hauser has a total of 24 co-authors. Here are the top co-authors by number of shared publications:
- Francisco-Jesús Castro-Jiménez
Universidad de Sevilla
4 shared publications - Christoph Koutschan
RICAM, Austrian Academy of Sciences, Linz, Austria
3 shared publications - Josef Schicho
Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences
3 shared publications - Florian Fürnsinn2 shared publications
M. E. Alonso
2 shared publications- Maria Emilia Alonso
Departamento de Álgebra Geometría y Topología Facultad de Ciencias Matemáticas Universidad Complutense Madrid Spain
2 shared publications
Latest Publications
- Open Access
- 7 November 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Publications of the Research Institute for Mathematical Sciences
- Vol. 60 (4) , 767-813
Resolving Surface Singularities in Positive Characteristic
The paper contains a systematic proof for the embedded resolution of two-dimensional hypersurface singularities over a field of arbitrary characteristic. It relies on the construction of a local upper semicontinuous invariant whose top locus defines at each stage of the resolution process the center of a permissible blowup under which the invariant improves. As the invariant belongs to a well-ordered set, resolution is achieved in finitely many steps. To define the invariant we analyze in detail the obstructions which occur when transcribing the inductive characteristic zero proof à la Hironaka. These difficulties are then overcome by introducing finer invariants than in zero characteristic: They reflect and control typical characteristic p phenomena. This may give new ideas for approaching the characteristic p resolution in higher dimensions.
- 7 November 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Publications of the Research Institute for Mathematical Sciences
- Vol. 60 (4) , 767-813
Open AccessResolving Surface Singularities in Positive Characteristic
The paper contains a systematic proof for the embedded resolution of two-dimensional hypersurface singularities over a field of arbitrary characteristic. It relies on the construction of a local upper semicontinuous invariant whose top locus defines at each stage of the resolution process the center of a permissible blowup under which the invariant improves. As the invariant belongs to a well-ordered set, resolution is achieved in finitely many steps. To define the invariant we analyze in detail the obstructions which occur when transcribing the inductive characteristic zero proof à la Hironaka. These difficulties are then overcome by introducing finer invariants than in zero characteristic: They reflect and control typical characteristic p phenomena. This may give new ideas for approaching the characteristic p resolution in higher dimensions.
- Open Access
On Abel's Problem about Logarithmic Integrals in Positive Characteristic
Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2, \ldots)(\!( x)\!)$ of $K(x)$, where $x'=1, z_1'=1/x$ and $z_i'=z_{i-1}'/z_{i-1}$. For differential equations of order $1$ it is shown that there exists a solution $y$ whose projections $y\vert_{z_{i+1}=z_{i+2}=\cdots=0}$ are algebraic over the field of rational functions $K(x, z_1, \ldots, z_{i})$ for all $i$. This can be seen as a characteristic $p$ analogue of Abel's problem about the algebraicity of logarithmic integrals. Further, the existence of infinite product representations of these solutions is shown. As a main tool $p^i$-curvatures are introduced, generalizing the notion of the $p$-curvature.Open AccessOn Abel's Problem about Logarithmic Integrals in Positive Characteristic
Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2, \ldots)(\!( x)\!)$ of $K(x)$, where $x'=1, z_1'=1/x$ and $z_i'=z_{i-1}'/z_{i-1}$. For differential equations of order $1$ it is shown that there exists a solution $y$ whose projections $y\vert_{z_{i+1}=z_{i+2}=\cdots=0}$ are algebraic over the field of rational functions $K(x, z_1, \ldots, z_{i})$ for all $i$. This can be seen as a characteristic $p$ analogue of Abel's problem about the algebraicity of logarithmic integrals. Further, the existence of infinite product representations of these solutions is shown. As a main tool $p^i$-curvatures are introduced, generalizing the notion of the $p$-curvature. - Open Access
Fuchs' theorem on linear differential equations in arbitrary characteristic
The paper generalizes Lazarus Fuchs' theorem on the solutions of complex ordinary linear differential equations with regular singularities to the case of ground fields of arbitrary characteristic, giving a precise description of the shape of each solution. This completes partial investigations started by Taira Honda and Bernard Dwork. The main features are the introduction of a differential ring $\mathcal{R}$ in infinitely many variables mimicking the role of the (complex) iterated logarithms, and the proof that adding these "logarithms" already provides sufficiently many primitives so as to solve any differential equation with regular singularity in $\mathcal{R}$. A key step in the proof is the reduction of the involved differential operator to an Euler operator, its normal form, to solve Euler equations in $\mathcal{R}$ and to lift their (monomial) solutions to solutions of the original equation. The first (and already very striking) example of this outset is the exponential function $\exp_p$ in positive characteristic, solution of $y' = y$. We prove that it necessarily involves all variables and we construct its explicit (and quite mysterious) power series expansion. Additionally, relations of our results to the Grothendieck-Katz $p$-curvature conjecture and related conjectures will be discussed.Open AccessFuchs' theorem on linear differential equations in arbitrary characteristic
The paper generalizes Lazarus Fuchs' theorem on the solutions of complex ordinary linear differential equations with regular singularities to the case of ground fields of arbitrary characteristic, giving a precise description of the shape of each solution. This completes partial investigations started by Taira Honda and Bernard Dwork. The main features are the introduction of a differential ring $\mathcal{R}$ in infinitely many variables mimicking the role of the (complex) iterated logarithms, and the proof that adding these "logarithms" already provides sufficiently many primitives so as to solve any differential equation with regular singularity in $\mathcal{R}$. A key step in the proof is the reduction of the involved differential operator to an Euler operator, its normal form, to solve Euler equations in $\mathcal{R}$ and to lift their (monomial) solutions to solutions of the original equation. The first (and already very striking) example of this outset is the exponential function $\exp_p$ in positive characteristic, solution of $y' = y$. We prove that it necessarily involves all variables and we construct its explicit (and quite mysterious) power series expansion. Additionally, relations of our results to the Grothendieck-Katz $p$-curvature conjecture and related conjectures will be discussed. - Open Access
On the formal neighborhood of a degenerate arc
We prove a result describing the structure of the formal neighborhoods of certain arcs in the arc space of an algebraic variety which are completely contained in the singular locus. In particular, we provide a precise formulation of the intuitive statement that constant arcs centered in the singular locus are the most singular points of the arc space.
Open AccessOn the formal neighborhood of a degenerate arc
We prove a result describing the structure of the formal neighborhoods of certain arcs in the arc space of an algebraic variety which are completely contained in the singular locus. In particular, we provide a precise formulation of the intuitive statement that constant arcs centered in the singular locus are the most singular points of the arc space.
- Open Access
- 1 December 2021
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 9
Arquile Varieties – Varieties Consisting of Power Series in a Single Variable
Spaces of power series solutions $y(\mathrm {t})$ in one variable $\mathrm {t}$ of systems of polynomial, algebraic, analytic or formal equations $f(\mathrm {t},\mathrm {y})=0$ can be viewed as ‘infinite-dimensional’ varieties over the ground field $\mathbf {k}$ as well as ‘finite-dimensional’ schemes over the power series ring $\mathbf {k}[[\mathrm {t}]]$ . We propose to call these solution spaces arquile varieties, as an enhancement of the concept of arc spaces. It will be proven that arquile varieties admit a natural stratification ${\mathcal Y}=\bigsqcup {\mathcal Y}_d$ , $d\in {\mathbb N}$ , such that each stratum ${\mathcal Y}_d$ is isomorphic to a Cartesian product ${\mathcal Z}_d\times \mathbb A^{\infty }_{\mathbf {k}}$ of a finite-dimensional, possibly singular variety ${\mathcal Z}_d$ over $\mathbf {k}$ with an affine space $\mathbb A^{\infty }_{\mathbf {k}}$ of infinite dimension. This shows that the singularities of the solution space of $f(\mathrm {t},\mathrm {y})=0$ are confined, up to the stratification, to the finite-dimensional part. Our results are established simultaneously for algebraic, convergent and formal power series, as well as convergent power series with prescribed radius of convergence. The key technical tool is a linearisation theorem, already used implicitly by Greenberg and Artin, showing that analytic maps between power series spaces can be essentially linearised by automorphisms of the source space. Instead of stratifying arquile varieties, one may alternatively consider formal neighbourhoods of their regular points and reprove with similar methods the Grinberg–Kazhdan–Drinfeld factorisation theorem for arc spaces in the classical setting and in the more general setting.- 1 December 2021
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 9
Open AccessArquile Varieties – Varieties Consisting of Power Series in a Single Variable
Spaces of power series solutions $y(\mathrm {t})$ in one variable $\mathrm {t}$ of systems of polynomial, algebraic, analytic or formal equations $f(\mathrm {t},\mathrm {y})=0$ can be viewed as ‘infinite-dimensional’ varieties over the ground field $\mathbf {k}$ as well as ‘finite-dimensional’ schemes over the power series ring $\mathbf {k}[[\mathrm {t}]]$ . We propose to call these solution spaces arquile varieties, as an enhancement of the concept of arc spaces. It will be proven that arquile varieties admit a natural stratification ${\mathcal Y}=\bigsqcup {\mathcal Y}_d$ , $d\in {\mathbb N}$ , such that each stratum ${\mathcal Y}_d$ is isomorphic to a Cartesian product ${\mathcal Z}_d\times \mathbb A^{\infty }_{\mathbf {k}}$ of a finite-dimensional, possibly singular variety ${\mathcal Z}_d$ over $\mathbf {k}$ with an affine space $\mathbb A^{\infty }_{\mathbf {k}}$ of infinite dimension. This shows that the singularities of the solution space of $f(\mathrm {t},\mathrm {y})=0$ are confined, up to the stratification, to the finite-dimensional part. Our results are established simultaneously for algebraic, convergent and formal power series, as well as convergent power series with prescribed radius of convergence. The key technical tool is a linearisation theorem, already used implicitly by Greenberg and Artin, showing that analytic maps between power series spaces can be essentially linearised by automorphisms of the source space. Instead of stratifying arquile varieties, one may alternatively consider formal neighbourhoods of their regular points and reprove with similar methods the Grinberg–Kazhdan–Drinfeld factorisation theorem for arc spaces in the classical setting and in the more general setting. - 21 October 2019
- Published by European Mathematical Society - EMS - Publishing House GmbH in Publications of the Research Institute for Mathematical Sciences
- Vol. 55 (4) , 835-857
Characterizing the Increase of the Residual Order under Blowup in Positive Characteristic
In contrast to the characteristic-zero situation, the residual order of an ideal may increase in positive characteristic under permissible blowups at points of the exceptional divisor where the order of the ideal has remained constant. The specific situations where this happens are described explicitly.
- 21 October 2019
- Published by European Mathematical Society - EMS - Publishing House GmbH in Publications of the Research Institute for Mathematical Sciences
- Vol. 55 (4) , 835-857
Characterizing the Increase of the Residual Order under Blowup in Positive Characteristic
In contrast to the characteristic-zero situation, the residual order of an ideal may increase in positive characteristic under permissible blowups at points of the exceptional divisor where the order of the ideal has remained constant. The specific situations where this happens are described explicitly.
- 4 January 2019
- Published by American Mathematical Society (AMS) in Journal of Algebraic Geometry
- Vol. 28 (2) , 391-403
Cycles of singularities appearing in the resolution problem in positive characteristic
We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It is not a counterexample to the resolution in positive characteristic since larger centers are also permissible and prevent the phenomenon from happening.
- 4 January 2019
- Published by American Mathematical Society (AMS) in Journal of Algebraic Geometry
- Vol. 28 (2) , 391-403
Cycles of singularities appearing in the resolution problem in positive characteristic
We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It is not a counterexample to the resolution in positive characteristic since larger centers are also permissible and prevent the phenomenon from happening.
- 18 May 2018
- Published by Wiley in Bulletin of the London Mathematical Society
- Vol. 50 (4) , 649-662
Echelons of power series and Gabrielov's counterexample to nested linear Artin approximation
- 18 May 2018
- Published by Wiley in Bulletin of the London Mathematical Society
- Vol. 50 (4) , 649-662
Echelons of power series and Gabrielov's counterexample to nested linear Artin approximation
- Open Access
Echelons of power series and Gabrielov's counterexample to nested linear Artin Approximation
Gabrielov's famous example for the failure of analytic Artin approximation in the presence of nested subring conditions is shown to be due to a growth phenomenon in standard basis computations for echelons, a generalization of the concept of ideals in power series rings.
Open AccessEchelons of power series and Gabrielov's counterexample to nested linear Artin Approximation
Gabrielov's famous example for the failure of analytic Artin approximation in the presence of nested subring conditions is shown to be due to a growth phenomenon in standard basis computations for echelons, a generalization of the concept of ideals in power series rings.