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104
Lifting smooth curves over invariants for representations of compact Lie groups
 TRANSFORMATION GROUPS
, 2000
"... We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption. ..."
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We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
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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Cited by 17 (4 self)
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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
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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Cited by 13 (3 self)
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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.
Topological lines in 3d tensor fields and discriminant hessian factorization
 IEEE Transactions on Visualization and Computer Graphics
, 2005
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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
Enumerative real algebraic geometry
 Algorithmic and Quantitative Real Algebraic Geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 60, AMS
, 2003
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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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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.
Real Solutions to Equations From Geometry
"... 1.1 Polyhedral bounds................................ 2 1.2 Upper bounds................................... 3 1.3 The Wronski map and the Shapiro Conjecture................. 5 ..."
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Cited by 10 (3 self)
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1.1 Polyhedral bounds................................ 2 1.2 Upper bounds................................... 3 1.3 The Wronski map and the Shapiro Conjecture................. 5
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.