Results 1  10
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12
PólyaSchur master theorems for circular domains and their boundaries
 Ann. of Math
"... Abstract. We characterize all linear operators on finite or infinitedimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Ω ⊆ C for arbitrary closed circular domains Ω (i.e., images of the closed unit disk under a Möbius transformation) and their ..."
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Cited by 22 (16 self)
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Abstract. We characterize all linear operators on finite or infinitedimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Ω ⊆ C for arbitrary closed circular domains Ω (i.e., images of the closed unit disk under a Möbius transformation) and their boundaries. This provides a natural framework for dealing with several longstanding fundamental problems, which we solve in a unified way. In particular, for Ω = R our results settle open questions that go back to Laguerre and PólyaSchur. 1.
Multivariate PólyaSchur classification problems in the Weyl algebra
, 2008
"... A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the higher dimensional generalizat ..."
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Cited by 12 (8 self)
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A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the higher dimensional generalization of PólyaSchur’s notion of multiplier sequence. We characterize all multivariate multiplier sequences as well as those of finite order. Next, we establish a multivariate extension of the CauchyPoincaré interlacing theorem and prove a natural analog of the Lax conjecture for real stable polynomials in two variables. Using the latter we describe all operators in A1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. Our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in An preserves stability if and only if its FischerFock adjoint does. These are powerful multivariate extensions of the classical HermitePoulainJensen theorem, Pólya’s curve theorem and SchurMalóSzegö composition theorems. Examples, applications to strict stability preservers and further directions are also discussed.
Classification of hyperbolicity and stability preservers: the multivariate Weyl algebra case
"... Abstract. A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We characterize all finite order linear differential operators that preserve stability. An important technical tool that we develop in the process is the multivariate generalizat ..."
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Cited by 9 (6 self)
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Abstract. A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We characterize all finite order linear differential operators that preserve stability. An important technical tool that we develop in the process is the multivariate generalization of the classical notion of multiplier sequence. We give a complete description of all multivariate multiplier sequences as well as those of finite order. Next we formulate and prove a natural analog of the Lax conjecture for real stable polynomials in two variables and use it to classify all finite order linear differential operators that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. As a further consequence of our methods we establish a duality theorem showing that a differential operator preserves stability if and only if its FischerFock adjoint has the same property. This is a vast generalization of the HermitePoulainJensen theorem in the univariate case and a natural multivariate extension of the latter. We also discuss several other applications of our results as well as further directions and open problems. Contents
fVECTORS OF BARYCENTRIC SUBDIVISIONS
"... Abstract. For a simplicial complex or more generally Boolean cell complex ∆ we study the behavior of the f and hvector under barycentric subdivision. We show that if ∆ has a nonnegative hvector then the hpolynomial of its barycentric subdivision has only simple and real zeros. As a consequence ..."
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Cited by 6 (1 self)
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Abstract. For a simplicial complex or more generally Boolean cell complex ∆ we study the behavior of the f and hvector under barycentric subdivision. We show that if ∆ has a nonnegative hvector then the hpolynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the CharneyDavis conjecture for spheres that are the subdivision of a Boolean cell complex or the subdivision of the boundary complex of a simple polytope. For a general (d − 1)dimensional simplicial complex ∆ the hpolynomial of its nth iterated subdivision shows convergent behavior. More precisely, we show that among the zeros of this hpolynomial there is one converging to infinity and the other d − 1 converge to a set of d − 1 real numbers which only depends on d. 1.
ACTIONS ON PERMUTATIONS AND UNIMODALITY OF DESCENT POLYNOMIALS
, 2006
"... We study an action (the SWGaction) on permutations due to Shapiro, Woan and Getu and use it to prove that the descent generating polynomial of certain sets of permutations has a nonnegative expansion in the basis {ti(1 + t) n−1−2i} ⌊(n−1)/2⌋ i=0. This property implies symmetry and unimodality. We ..."
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Cited by 5 (0 self)
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We study an action (the SWGaction) on permutations due to Shapiro, Woan and Getu and use it to prove that the descent generating polynomial of certain sets of permutations has a nonnegative expansion in the basis {ti(1 + t) n−1−2i} ⌊(n−1)/2⌋ i=0. This property implies symmetry and unimodality. We prove that the SWGaction is invariant under stacksorting which strengthens recent unimodality results of Bóna. We prove that the generalized permutation patterns (13 2) and (2 31) are invariant under the SWGaction and use this to prove unimodality properties for a qanalog of the Eulerian numbers recently studied by Corteel, Postnikov, Steingrímsson and Williams. We also generalize the SWGaction to linear extensions of signgraded posets to give a new proof of the unimodality of the (P, ω)Eulerian polynomials and a combinatorial interpretations (in terms of Stembridge’s peak polynomials) of the corresponding coefficients when expanded in the above basis. Finally, we prove that the statistic defined as the number of vertices of even height in the unordered decreasing tree of a permutation has the same distribution as the number of descents on any set of permutations invariant under the SWGaction. When restricted to the stacksortable permutations we recover a result of Kreweras.
qEulerian polynomials and polynomials with only real zeros, Electron
 J. Combin. 15 (2008), Research Paper
"... Let f and F be two polynomials satisfying F (x) = u(x)f(x) + v(x)f ′ (x). We characterize the relation between the location and multiplicity of the real zeros of f and F, which generalizes and unifies many known results, including the results of Brenti and Brändén about the qEulerian polynomials. ..."
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Cited by 1 (1 self)
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Let f and F be two polynomials satisfying F (x) = u(x)f(x) + v(x)f ′ (x). We characterize the relation between the location and multiplicity of the real zeros of f and F, which generalizes and unifies many known results, including the results of Brenti and Brändén about the qEulerian polynomials. 1
THE γVECTOR OF A BARYCENTRIC SUBDIVISION
, 1003
"... Abstract. We prove that the γvector of the barycentric subdivision of a simplicial sphere is the fvector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the hvector of the barycen ..."
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Abstract. We prove that the γvector of the barycentric subdivision of a simplicial sphere is the fvector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the hvector of the barycentric subdivision of a boolean complex. 1.
POLYNOMIAL SEQUENCES WITH ONLY REAL ZEROS
, 2005
"... Abstract. In this paper we develop methods to provide unified approaches to the reality of zeros of polynomial sequences satisfying certain recurrence relations and then apply them to derive some known results and solve certain open problems and conjectures. 1. ..."
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Abstract. In this paper we develop methods to provide unified approaches to the reality of zeros of polynomial sequences satisfying certain recurrence relations and then apply them to derive some known results and solve certain open problems and conjectures. 1.