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Let $(X, 0)$ be a reduced, equidimensional germ of an analytic singularity with reduced tangent cone $(C_{X, 0}, 0)$ . We prove that the absence of exceptional cones is a necessary and sufficient condition for the smooth part $𝔛^{0}$ of the specialization to the tangent cone $ϕ : 𝔛 \to ℂ$ to satisfy Whitney’s conditions along the parameter axis $Y$ . This result is a first step in generalizing to higher dimensions Lê and Teissier’s result for hypersurfaces of $ℂ^{3}$ which establishes the Whitney equisingularity of $X$ and its tangent cone under...

Specializations of one-parameter families of polynomials

Farshid Hajir, Siman Wong (2006)

Annales de l’institut Fourier

Let $K$ be a number field, and suppose $λ (x, t) \in K [x, t]$ is irreducible over $K (t)$ . Using algebraic geometry and group theory, we describe conditions under which the $K$ -exceptional set of $λ$ , i.e. the set of $α \in K$ for which the specialized polynomial $λ (x, α)$ is $K$ -reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed $n \geq 10$ , all but finitely many $K$ -specializations of the degree $n$ generalized Laguerre polynomial $L_{n}^{(t)} (x)$ are $K$ -irreducible and have Galois group $S_{n}$ . Second, we study specializations...

Spectra of differential rings

William F. Keigher (1983)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Spectral curves and integrable systems

Eyal Markman (1994)

Compositio Mathematica

Spectral curves and the generalised theta divisor.

S. Ramanan, M.S. Narasimhan, A. Beauville (1989)

Journal für die reine und angewandte Mathematik

Spectral curves of operators with elliptic coefficients.

Eilbeck, J.Chris, Enolski, Victor Z., Previato, Emma (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Spectral curves, opers and integrable systems

David Ben-Zvi, Edward Frenkel (2001)

Publications Mathématiques de l'IHÉS

Spectral geometry of semi-algebraic sets

Mikhael Gromov (1992)

Annales de l'institut Fourier

The spectrum of the Laplace operator on algebraic and semialgebraic subsets $A$ in $R^{N}$ is studied and the number of small eigenvalues is estimated by the degree of $A$ .

Spectral Pairs, Mixed Hodge Modules, and Series of Plane Curve Singularities.

Nemethi, A., Steenbrink, J.H.M. (1995)

The New York Journal of Mathematics [electronic only]

Spectral Real Semigroups

M. Dickmann, A. Petrovich (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

The notion of a real semigroup was introduced in [8] to provide a framework for the investigation of the theory of (diagonal) quadratic forms over commutative, unitary, semi-real rings. In this paper we introduce and study an outstanding class of such structures, that we call spectral real semigroups (SRS). Our main results are: (i) The existence of a natural functorial duality between the category of SRSs and that of hereditarily normal spectral spaces; (ii) Characterization of the SRSs as the...

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