Search results for "algebra"

Rational quasi-projective surfaces with algebraic moduli of real forms

2022

We construct real rational quasi-projective surfaces with positive dimensional algebraic moduli of mutually non-isomorphic real forms.

On the projective geometry of entanglement and contextuality

2019

$\mathbb{A}^1$-cylinders over smooth affine surfaces of negative Kodaira dimension

2019

International audience; The Zariski Cancellation problem for smooth affine surfaces asks whether two suchsurfaces whose products with the affine line are isomorphic are isomorphic themselves. Byresults of Iitaka-Fujita, the answer is positive for surfaces of non-negative Kodaira dimen-sion. By a characterization due to Miyanishi, surfaces of negative Kodaira dimension arefibered by the affine line, and by a celebrated result of Miyanishi-Sugie, the answer to theproblem is positive if one of the surfaces is the affine plane. On the other hand, exam-ples of non-isomorphicA1-fibered affine surfaces with isomorphicA1-cylinders were firstconstructed by Danielewski in 1989, and then by many other…

Stable motivic homotopy theory at infinity

2021

In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under $\ell$-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at in…

A Symplectic Kovacic's Algorithm in Dimension 4

2018

Let $L$ be a $4$th order differential operator with coefficients in $\mathbb{K}(z)$, with $\mathbb{K}$ a computable algebraically closed field. The operator $L$ is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions $X$ satisfies $X^t J X=J$ where $J$ is the standard symplectic matrix. It is called projectively symplectic when it is projectively equivalent to a symplectic operator. We design an algorithm to test if $L$ is projectively symplectic. Furthermore, based on Kovacic's algorithm, we design an algorithm that computes Liouvillian solutions of projectively symplectic operators of order $4$. Moreover, using Klein's Theorem, algebraic solution…

Quantifier elimination in the quasi-analytic framework

2012

We associate to every compact polydisk B [belonging to ] Rn an algebra CB of real functions defined in a neighborhood of B. The collection of these algebras is supposed to be closed under several operations, such as composition and partial derivatives. Moreover, if the center of B is the origin, we assume that the algebra of germs at the origin of elements of CB is quasianalytic (it does not contain any flat germ). We define with these functions the collection of C-semianalytic and C-subanalytic sets according to the classical process in real analytic geometry. Our main result is an analogue of Tarski-Seidenberg's usual result for these sets. It says that the sub-C-subanalytic sets may be d…

Local monomialization of generalized real analytic functions

2011

Generalized power series extend the notion of formal power series by considering exponents ofeach variable ranging in a well ordered set of positive real numbers. Generalized analytic functionsare defined locally by the sum of convergent generalized power series with real coe cients. Weprove a local monomialization result for these functions: they can be transformed into a monomialvia a locally finite collection of finite sequences of local blowingsup. For a convenient frameworkwhere this result can be established, we introduce the notion of generalized analytic manifoldand the correct definition of blowing-up in this category.

Geometrical construction of quantum groups representations

2002

We describe geometrically the classical and quantum inhomogeneous groups $G_0=(SL(2, \BbbC)\triangleright \BbbC^2)$ and $G_1=(SL(2, \BbbC)\triangleright \BbbC^2)\triangleright \BbbC$ by studying explicitly their shape algebras as a spaces of polynomial functions with a quadratic relations.

Kontsevich and Takhtajan construction of star product on the Poisson Lie group GL(2)

2001

Comparing the star product defined by Takhtajan on the Poisson-Lie group GL(2) and any star product calculated from the Kontsevich's graphs (any ''K-star product'') on the same group, we show, by direct computation, that the Takhtajan star product on GL(2) can't be written as a K-star product.

A possible quantic motivation of the structure of quantum group: continuation

2012

Motivated by Quantum Mechanics considerations, we expose some cross product constructions on a groupoid structure. Furthermore, critical remarks are made on some basic formal aspects of the Hopf algebra structure.