Algebraic boundaries of convex semi-algebraic sets

Sinn, Rainer

doi:10.1186/s40687-015-0022-0

Cited by 22 publications

(27 citation statements)

References 14 publications

(22 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For similar treatment, see [35]. The following algorithm has the same input as Algorithm 3.1 and returns a finite sequence of (Φ k , Z k ) with the property that for any γ ∈ dom(c * 0 (c | C h )), there exists a k, such that if Z k (γ) = 0, then c * 0 is contained in the roots of Φ k (c0, γ).…”

Section: Example 32 Consider the Astroid Which Is A Real Locus Of Amentioning

confidence: 97%

Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety

Guo

Din

Wang

et al. 2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

International audienceWe consider the problem of optimizing a parametric linear function over a non-compact real trace of an algebraic set. Our goal is to compute a representing polynomial which defines a hypersurface containing the graph of the optimal value function. Rostalski and Sturmfels showed that when the algebraic set is irreducible and smooth with a compact real trace, then the least degree representing polynomial is given by the defining polynomial of the irreducible hypersurface dual to the projective closure of the algebraic set.First, we generalize this approach to non-compact situations. We prove that the graph of the opposite of the optimal value function is still contained in the affine cone over a dual variety similar to the one considered in compact case. In consequence, we present an algorithm for solving the considered parametric optimization problem for generic parameters' values. For some special parameters' values, the representing polynomials of the dual variety can be identically zero, which give no information on the optimal value. We design a dedicated algorithm that identifies those regions of the parameters' space and computes for each of these regions a new polynomial defining the optimal value over the considered region

show abstract

Section: Example 32 Consider the Astroid Which Is A Real Locus Of Amentioning

confidence: 97%

Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety

Guo

Din

Wang

et al. 2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

show abstract

“…In the following propositions in this section, we will prove the following theorem. 3,6,9,12,13,12,9,6,3,1), 3,6,10,13,14,13,10,6,3,1), 3,6,10,14,15,14,10,6,3,1), 3,6,10,14,16,14,10,6,3,1), 3,6,10,15,17,15,10,6,3,1).…”

Section: Extreme Rays Of Maximal Rank and Positive Gorenstein Idealsmentioning

confidence: 99%

“…The construction given in the preceding section for extreme rays of maximal rank d+2 2 −4 leads to an extreme ray R + ℓ of Σ ∨ 10 such that the Hilbert function of the corresponding Gorenstein ideal I(ℓ) is 3,6,10,15,17,15,10,6,3,1).…”

Section: Extreme Rays Of Maximal Rank and Positive Gorenstein Idealsmentioning

confidence: 99%

Extreme rays of Hankel spectrahedra for ternary forms

Blekherman

Sinn

2017

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

The cone of sums of squares is one of the central objects in convex algebraic geometry. Its defining linear inequalities correspond to the extreme rays of the dual convex cone. This dual cone is a spectrahedron, which can be explicitly realized as a section of the cone of positive semidefinite matrices with the linear subspace of Hankel (or middle catalecticant) matrices. In this paper we initiate a systematic study of the extreme rays of Hankel spectrahedra for ternary forms. We show that the Zariski closure of the union of extreme rays is the variety of all Hankel matrices of corank at least 4, an irreducible variety of codimension 10 and we determine its degree. We explicitly construct an extreme ray of maximal rank using the Cayley-Bacharach Theorem for plane curves. We apply our results to the study of the algebraic boundary of the cone of sums of squares. Its irreducible components are dual varieties to varieties of Gorenstein ideals with certain Hilbert functions. We determine these Hilbert functions for some cases of small degree. We also observe surprising gaps in the ranks of Hankel matrices of the extreme rays.We work out the first three nontrivial cases d = 3, 4, 5 in Section 3, extending the study of the algebraic boundary of the sums of squares cones for ternary sextics and quaternary quartics in [4]. More specifically we show in section 3:Proposition. The Hankel spectrahedron Σ ∨ 8 has extreme rays of rank 1, 10, and 11. We construct extreme rays of rank 10 and 11 such that the Hilbert

show abstract

“…There is a slight twist to using the codimension 1 case above as the induction hypothesis because π(Q) is generally not a spectrahedron. However, it is closed, convex and semialgebraic, so ∂ a π(Q) is also defined by a polynomial f , see [12,Lemma 2.5]. Now let π : R n → R d be a generic projection with n − d > 1.…”

Section: One Proof and Many Numbersmentioning

confidence: 99%

“…The boundary ∂S of a non-empty generic spectrahedral shadow S is a semi-algebraic set of codimension 1 in R d ; see [12,Corollary 2.6]. Up to scaling, there exists a unique squarefree polynomial Φ S ∈ R[x 1 , .…”

Section: Introductionmentioning

confidence: 99%