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Elementary proof of the B. and M. Shapiro conjecture for rational functions
, 2005
"... We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation φ such that φ ◦ g is a real rational function. Then we interpret the result in terms of Fuchsian differential equation ..."
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Cited by 7 (5 self)
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We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation φ such that φ ◦ g is a real rational function. Then we interpret the result in terms of Fuchsian differential equations whose general solution is a polynomial and in terms of electrostatics.
Solving Schubert problems with LittlewoodRichardson homotopies
 Proc. ISSAC 2010
"... Abstract. We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This LittlewoodRichardson homotopy is based on Vakil’s geometric proof of the LittlewoodRichardson rule. Its start solutions are given by linear equations and they ..."
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Cited by 6 (5 self)
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Abstract. We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This LittlewoodRichardson homotopy is based on Vakil’s geometric proof of the LittlewoodRichardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The LittlewoodRichardson homotopy algorithm is implemented using the path trackers of the software package PHCpack. 1.
Real aspects of the moduli space of stable maps of genus zero curves
 TURKISH JOURNAL OF MATHEMATICS, VOL
, 2003
"... We show that the moduli space of stable maps from a genus 0 curve into a nonsingular real convex projective variety having a real structure compatible with a complex conjugate involution on CP k has a real structure. The real part of this moduli space consists of real maps having marked points on ..."
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Cited by 4 (2 self)
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We show that the moduli space of stable maps from a genus 0 curve into a nonsingular real convex projective variety having a real structure compatible with a complex conjugate involution on CP k has a real structure. The real part of this moduli space consists of real maps having marked points on the real part of domain curves. This real part analysis enables us to relate the studies of real intersection cycles with real enumerative problems.
Transversality properties on the moduli space of genus 0 stable maps to CP 2 blownup at finte points
"... Abstract. We characterize transversality, nontransversality properties on the moduli space of genus 0 stable maps to a rational projective surface. This characterization is based on the singularity analysis of the product of evaluation maps and deformation properties of the pointed stable maps. If ..."
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Cited by 3 (1 self)
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Abstract. We characterize transversality, nontransversality properties on the moduli space of genus 0 stable maps to a rational projective surface. This characterization is based on the singularity analysis of the product of evaluation maps and deformation properties of the pointed stable maps. If a target space is equipped with a real structure, i.e, antiholomorphic involution, then this transversality result has a real enumerative implication. One of the natural consequence of this result is Welschinger’s invariant in an algebraic geometry category, which is about the global minimum bound of the number of real stable maps passing through certain real points in a general position on the target space. 1.
Real lines tangent to 2n−2 quadrics
 in R n
"... Abstract. We show that for each n ≥ 2 there are 1 n22n−2 () ..."
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Cited by 2 (2 self)
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Abstract. We show that for each n ≥ 2 there are 1 n22n−2 ()
Dedicated to the originator
, 2007
"... Real GromovWitten invariants on the moduli space of genus 0 stable maps to a smooth rational projective space ..."
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Real GromovWitten invariants on the moduli space of genus 0 stable maps to a smooth rational projective space
REAL ASPECTS OF THE MODULI SPACE OF GENUS ZERO STABLE MAPS
, 2007
"... Abstract. We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space. 1. ..."
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Abstract. We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space. 1.