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53
Frontiers of reality in Schubert calculus
 Bulletin of the AMS
"... Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fu ..."
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Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifications in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and generalizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.
Topological lines in 3D tensor fields and discriminant hessian factorization
 IEEE Transactions on Visualization and Computer Graphics
, 2005
"... Abstract — This paper addresses several issues related to topological analysis of 3D second order symmetric tensor fields. First, we show that the degenerate features in such data sets form stable topological lines rather than points as previously thought. Secondly, the paper presents two different ..."
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Abstract — This paper addresses several issues related to topological analysis of 3D second order symmetric tensor fields. First, we show that the degenerate features in such data sets form stable topological lines rather than points as previously thought. Secondly, the paper presents two different methods for extracting these features by identifying the individual points on these lines and connecting them. Thirdly, this paper proposes an analytical form of obtaining tangents at the degenerate points along these topological lines. The tangents are derived from a Hessian factorization technique on the tensor discriminant and leads to a fast and stable solution. Together, these three advances allow us to extract the backbone topological lines that form the basis for topological analysis of tensor fields. Index terms: hyperstreamlines, real symmetric tensors, degenerate tensors, tensor topology. I.
LMI approximations for cones of positive semidefinite forms
 Fachbereich Mathematik, Universität Konstanz, 78457
"... An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequen ..."
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Cited by 8 (2 self)
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An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositive matrices, and to certain optimization problems involving random variables with some known moment information. We bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms (homogeneous polynomials). Our approach enables us to extend the existing methodology to new approximation schemes. In particular, we derive a novel approximation to the cone of copositive forms, that is, the cone of forms that are positive semidefinite over the nonnegative orthant. The format of our construction can be extended to forms that are positive semidefinite over more general conic domains. We also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain. This opens the possibility of using linear programming technology in optimization problems over these cones.
Convexity properties of the cone of nonnegative polynomials
 Discrete Comput. Geom
"... We study metric properties of the cone of homogeneous nonnegative multivariate polynomials and the cone of sums of powers of linear forms, and the relationship between the two cones. We compute the maximum volume ellipsoid of the natural base of the cone of nonnegative polynomials and the minimum v ..."
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We study metric properties of the cone of homogeneous nonnegative multivariate polynomials and the cone of sums of powers of linear forms, and the relationship between the two cones. We compute the maximum volume ellipsoid of the natural base of the cone of nonnegative polynomials and the minimum volume ellipsoid of the natural base of the cone of powers of linear forms and compute the coefficients of symmetry of the bases. The multiplication by (x 2 1 +... + x2 n) m induces an isometric embedding of the space of polynomials of degree 2k into the space of polynomials of degree 2(k + m), which allows us to compare the cone of nonnegative polynomials of degree 2k and the cone of sums of 2(k + m)powers of linear forms. We estimate the volume ratio of the bases of the two cones and the rate at which it approaches 1 as m grows.
A convex polynomial that is not sosconvex
 Mathematical Programming
"... A multivariate polynomial p(x) = p(x1,...,xn) is sosconvex if its Hessian H(x) can be factored as H(x) = M T (x)M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sosconvexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sosconvex ..."
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A multivariate polynomial p(x) = p(x1,...,xn) is sosconvex if its Hessian H(x) can be factored as H(x) = M T (x)M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sosconvexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sosconvexity of a polynomial can be cast as the feasibility of a semidefinite program, which can be solved efficiently. Motivated by this computational tractability, it has been recently speculated whether sosconvexity is also a necessary condition for convexity of polynomials. In this paper, we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sosconvex. Interestingly, our example is found with software using sum of squares programming techniques and the duality theory of semidefinite optimization. As a byproduct of our numerical procedure, we obtain a simple method for searching over a restricted family of nonnegative polynomials that are not sums of squares. 1
On Hilbert’s construction of positive polynomials
"... Abstract. In 1888, Hilbert described how to find real polynomials which take only nonnegative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert’s construction and present ..."
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Abstract. In 1888, Hilbert described how to find real polynomials which take only nonnegative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert’s construction and present many such polynomials. 1. History and Overview A real polynomial f(x1,...,xn) is psd or positive if f(a) ≥ 0 for all a ∈ R n; it is sos or a sum of squares if there exist real polynomials hj so that f = ∑ h 2 j. For forms, we follow the notation of [4] and use Pn,m to denote the cone of real psd forms of even degree m in n variables, Σn,m to denote its subcone of sos forms and let ∆n,m = Pn,m � Σn,m. The Fundamental Theorem of Algebra implies that ∆2,m = ∅; ∆n,2 = ∅ follows from the diagonalization of psd quadratic forms. The first suggestion that a psd form might not be sos was made by Minkowski in the oral defense of his 1885 doctoral dissertation: Minkowski proposed the thesis that not every psd form is sos. Hilbert was one of his official “opponents ” and remarked that Minkowski’s arguments had convinced him that this thesis should be true for ternary forms. (See [14], [15] and [24].) Three years later, in a single remarkable paper, Hilbert [11] resolved the question. He first showed that F ∈ P3,4 is a sum of three squares of quadratic forms; see [23] and [26] for recent expositions and [17, 18] for another approach. Hilbert then described a construction of forms in ∆3,6 and ∆4,4; after multiplying these by powers of linear forms if necessary, it follows that ∆n,m ̸ = ∅ if n ≥ 3 and m ≥ 6 or n ≥ 4 and m ≥ 4. The goal of this paper is to isolate the underlying mechanism of Hilbert’s construction, show that it applies to situations more general than those in [11], and use it to produce many new examples. In [11], Hilbert first worked with polynomials in two variables, which homogenize to ternary forms. Suppose f1(x, y) and f2(x, y) are two relatively prime real cubic polynomials with nine distinct real common zeros – {πi}, indexed arbitrarily – so that
POLYNOMIALS NONNEGATIVE ON A STRIP
"... Abstract. We prove that if f(x, y) is a polynomial with real coefficients which is nonnegative on the strip [0, 1] × R, then f(x, y) has a presentation of the form k∑ f(x, y) = gi(x, y) 2 ℓ∑ + hj(x, y) 2 x(1 − x), i=1 j=1 where the gi(x, y) and hj(x, y) are polynomials with real coefficients. 1. ..."
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Abstract. We prove that if f(x, y) is a polynomial with real coefficients which is nonnegative on the strip [0, 1] × R, then f(x, y) has a presentation of the form k∑ f(x, y) = gi(x, y) 2 ℓ∑ + hj(x, y) 2 x(1 − x), i=1 j=1 where the gi(x, y) and hj(x, y) are polynomials with real coefficients. 1.
LOWER BOUNDS FOR A POLYNOMIAL IN TERMS OF ITS COEFFICIENTS
"... Abstract. We determine new sufficient conditions in terms of the coefficients for a polynomial f ∈ R[X] of degree 2d (d ≥ 1) in n ≥ 1 variables to be a sum of squares of polynomials, thereby strengthening results of Fidalgo and Kovacec [2] and of Lasserre [6]. Exploiting these results, we determine, ..."
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Abstract. We determine new sufficient conditions in terms of the coefficients for a polynomial f ∈ R[X] of degree 2d (d ≥ 1) in n ≥ 1 variables to be a sum of squares of polynomials, thereby strengthening results of Fidalgo and Kovacec [2] and of Lasserre [6]. Exploiting these results, we determine, for any polynomial f ∈ R[X] of degree 2d whose highest degree term is an interior point in the cone of sums of squares of forms of degree d, a real number r such that f − r is a sum of squares of polynomials. The existence of such a number r was proved earlier by Marshall [8], but no estimates for r were given. We also determine a lower bound for any polynomial f whose highest degree term is positive definite. 1.
Positive Extensions And RieszFejer Factorization For TwoVariable Trigonometric Polynomials
"... In this paper the autoregressive filter problem for bivariate stochastic processes is reduced to a finite positive definite matrix completion problem where the completion is required to satisfy additional low rank conditions. The autoregressive filter problem may also be interpreted as a twovariabl ..."
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In this paper the autoregressive filter problem for bivariate stochastic processes is reduced to a finite positive definite matrix completion problem where the completion is required to satisfy additional low rank conditions. The autoregressive filter problem may also be interpreted as a twovariable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. For the proof a specific twovariable Kronecker theorem is developed, as well as a twovariable Christo#elDarboux formula. As a corollary of the main result a necessary and su#cient condition for the existence of a spectral RieszFejer factorization of a twovariable trigonometric polynomial is given in terms of the Fourier coe#cients of its reciprocal. Finally, numerical results are presented for both the autoregressive filter problem as well as the factorization problem. 1