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We present a new method to compute normal forms, applied to the germs of reversible mappings. We translate the classification problem of these germs to the theory of ideals in the space of the coefficients of their jets. Integral factorization coupled with Gröbner basis constructionjs the key factor that makes the process efficient. We also show that a language with typed objects like AXIOM is very convenient to solve these kinds of problems.

Effective base point freeness.

János Kollár (1993)

Mathematische Annalen

Effective bounds for Faltings’s delta function

Jay Jorgenson, Jürg Kramer (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces $X$ , nowadays called Faltings’s delta function and here denoted by $δ_{Fal} (X)$ . For a given compact Riemann surface $X$ of genus $g_{X} = g$ , the invariant $δ_{Fal} (X)$ is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space $ℳ_{g}$ of genus $g$ curves determined by $X$ to its boundary $\partial ℳ_{g}$ . In this paper we begin by revisiting a formula derived in [14],...

Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields

Hélène Esnault, Eckart Viehweg (1990)

Compositio Mathematica

Effective Convergence Bounds for Frobenius Structures on Connections

Kiran S. Kedlaya, Jan Tuitman (2012)

Rendiconti del Seminario Matematico della Università di Padova

Effective diophantine approximation on $𝔾_{M}$ , II

E. Bombieri, P. B. Cohen (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Effective divisors of higher rank on a curve and the Siegel formula

F. Ghione, M. Letizia (1992)

Compositio Mathematica

Effective equidistribution of S-integral points on symmetric varieties

Yves Benoist, Hee Oh (2012)

Annales de l’institut Fourier

Let $K$ be a global field of characteristic not 2. Let $Z = H ∖ G$ be a symmetric variety defined over $K$ and $S$ a finite set of places of $K$ . We obtain counting and equidistribution results for the S-integral points of $Z$ . Our results are effective when $K$ is a number field.

Effective Estimates for Unirationality.

Lorenzo Ramero (1990)

Manuscripta mathematica

Effective finite generation for adjoint rings

Paolo Cascini, De-Qi Zhang (2014)

Annales de l’institut Fourier

We describe a bound on the degree of the generators for some adjoint rings on surfaces and threefolds.

Effective freeness and point separation for adjoint bundles.

Yum-Tong Siu, Urban Angehrn (1995)

Inventiones mathematicae

Effective freeness of adjoint line bundles.

Heier, Gordon (2002)

Documenta Mathematica

Effective nonvanishing, effective global generation

Mark Andrea A. De Cataldo (1998)

Annales de l'institut Fourier

We prove a multiple-points higher-jets nonvanishing theorem by the use of local Seshadri constants. Applications are given to effectivity problems such as constructing rational and birational maps into Grassmannians, and the global generation of vector bundles.

Effective Nullstellensatz and geometric degree for zero-dimensional ideals

A. Fabianom, G. Pucci, A. Yger (1996)

Acta Arithmetica

Effective Nullstellensatz for arbitrary ideals

János Kollár (1999)

Journal of the European Mathematical Society

Let $f_{i}$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_{i}$ such that $\sum g_{i} f_{i} = 1$ . The effective versions of this result bound the degrees of the $g_{i}$ in terms of the degrees of the $f_{j}$ . The aim of this paper is to generalize this to the case when the $f_{i}$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.

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