Plane Algebraic Curves

Fischer, Gerd

doi:10.1090/stml/015

Cited by 107 publications

(117 citation statements)

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“…Now that we have confined S to R we will show that it consists of a single, smooth convex oval. The proof uses the idea of the Hessian curve associated to a plane algebraic curve [17]. For a curve in the projective plane defined by a homogeneous polynomial in three variables, the Hessian curve is the zero-set of the usual 3 × 3 Hessian determinant of second partial derivatives.…”

mentioning

confidence: 99%

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

Corbera

Cors

Llibre

et al. 2015

SIAM J. Math. Anal.

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Abstract. We study the relative equilibria of the limit case of the planar Newtonian 4-body problem when three masses tend to zero, the so-called (1+3)-body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1+3)-body problem can be continued to a unique family of relative equilibria of the general 4-body problem when three of the masses are sufficiently small and every convex relative equilibrium for these masses belongs to one of these six families.

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confidence: 99%

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

Corbera

Cors

Llibre

et al. 2015

SIAM J. Math. Anal.

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“…The tangent at a flex of an algebraic curve C is a non-ordinary singularity of the envelope of C (see [6], 5.4, for instance). Therefore, proposition 2.4 asserts in particular that (the one-dimensional part of) S has no flexes.…”

Section: Groups Of Pointsmentioning

confidence: 99%

“…Proof. Since C contains no line, there is a one to one correspondence γ → γ * , between the sets of branches of C and its envelope C * , so that if γ has origin p, tangent T , order n and classn, then γ * has origin T , tangent p * , ordern and class n (see for instance [13] or [6]). Furthermore, by definition, the number of times a line is counted among the tangents to C through a point p is the multiplicity of intersection [p * · C * ] .…”

Section: S(t))mentioning

confidence: 99%

Siebeck curves and two refinements of the

Casas-Alvero¹

2012

MATH. SCAND.

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“…The main interest in this condition lies in the fact that, contrary to conditions (2) and (3), it is not possible to use it to obtain information on A and B based only on the local behavior of the exponential around 0. Bourgeois showed that Condition (4) implies that A and B are simultaneously triangularizable if n = 2, and produced a proof that this also holds when n = 3.…”

Section: The Problemmentioning

confidence: 99%

On commuting matrices and exponentials

Pazzis¹

2012

Proc. Amer. Math. Soc.

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Let A and B be matrices of M n (C). We show that if exp(A) k exp(B) l = exp(kA + lB) for all integers k and l, then AB = BA. We also show that if exp(A) k exp(B) = exp(B) exp(A) k = exp(kA+B) for every positive integer k, then the pair (A, B) has property L of Motzkin and Taussky. As a consequence, if G is a subgroup of (M n (C), +) and M → exp(M ) is a homomorphism from G to (GL n (C), ×), then G consists of commuting matrices. If S is a subsemigroup of (M n (C), +) and M → exp(M ) is a homomorphism from S to (GL n (C), ×), then the linear subspace Span(S) of M n (C) has property L of Motzkin and Taussky.

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Plane Algebraic Curves

Cited by 107 publications

References 0 publications

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

Siebeck curves and two refinements of the

On commuting matrices and exponentials

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