Real Algebraic Geometry

Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise

doi:10.1007/978-3-662-03718-8

Cited by 1,474 publications

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“…[27]). We compute a bound for the diameter of the inner complement component using the upper SDP for the intervals [−3, −9] and [1,7] with steplength 0.1. For any of these points we compute 14 SDPs in order to estimate the radius (based on binary search).…”

Section: Computing the Relaxations Via Semidefinite Programmingmentioning

confidence: 99%

Approximating amoebas and coamoebas by sums of squares

Theobald

Wolff

2014

Math. Comp.

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Abstract. Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming.Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstellensatz, this allows to tackle the problem by sums of squares techniques and semidefinite programming. Our method yields polynomial identities as certificates of non-containment of a point in an amoeba or coamoeba. As the main theoretical result, we establish some degree bounds on the polynomial certificates. Moreover, we provide some actual computations of amoebas based on the sums of squares approach.

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Section: Computing the Relaxations Via Semidefinite Programmingmentioning

confidence: 99%

Approximating amoebas and coamoebas by sums of squares

Theobald

Wolff

2014

Math. Comp.

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“…[5]. Indeed, note that X can be embedded into some R N as a semialgebraic set and that the distance function f is a continuous semialgebraic function on X.…”

Section: Topological Preliminariesmentioning

confidence: 99%

Systolic invariants of groups and 2-complexes via Grushko decomposition

Rudyak¹,

Sabourau²

2008

Ann. inst. Fourier

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We prove a finiteness result for the systolic area of groups, answering a question of M. Gromov. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of 2-complexes whose systolic area is uniformly bounded. Furthermore, we prove a uniform systolic inequality for all 2-complexes with unfree fundamental group that improves the previously known bounds in this dimension.

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“…The ring NðV Þ is noetherian by [5,Theorem 8.7.18]. If m is a maximal ideal of NðV Þ, then m is the ideal of Nash functions vanishing at a point in M by [5,Corollary 8.6.3] and NðV Þ m is a regular local ring by [5,Proposition 8.7.15].…”

Section: The Nash Casementioning

confidence: 99%

“…If m is a maximal ideal of NðV Þ, then m is the ideal of Nash functions vanishing at a point in M by [5,Corollary 8.6.3] and NðV Þ m is a regular local ring by [5,Proposition 8.7.15]. So NðV Þ p is a regular local ring for any prime ideal p of NðV Þ, i.e.…”

Section: The Nash Casementioning

confidence: 99%

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Description of algebraically constructible functions

Bonnard

2003

Advg

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Abstract. The algebraically constructible functions on a real algebraic set are the sums of signs of polynomials on this set. We prove a formula giving the minimal number of polynomials needed to write generically a given algebraically constructible function as a sum of signs. We also prove a characterization of the polynomials appearing in a generic presentation of the function with the minimal number of polynomials. Both results are e¤ective.

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Real Algebraic Geometry

Abstract: The present edition is a substantially revised and expanded English version of the book "Geometrie algebrique reelle", originally published in French as Ergebnisse der Mathematik und ihrer

Cited by 1,474 publications

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Approximating amoebas and coamoebas by sums of squares

Approximating amoebas and coamoebas by sums of squares

Systolic invariants of groups and 2-complexes via Grushko decomposition

Description of algebraically constructible functions

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