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Convex geometry of orbits
 Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ
, 2005
"... Abstract. We study metric properties of convex bodies B and their polars B ◦ , where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G = Sn, the symmetric group), the set of nonnegative polynomials ..."
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Abstract. We study metric properties of convex bodies B and their polars B ◦ , where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G = Sn, the symmetric group), the set of nonnegative polynomials in real algebraic geometry (G = SO(n), the special orthogonal group), and the convex hull of the Grassmannian and the unit comass ball in the theory of calibrated geometries (G = SO(n), but with a different action). We compute the radius of the largest ball contained in the symmetric Traveling Salesman Polytope, give a reasonably tight estimate for the radius of the Euclidean ball containing the unit comass ball and review (sometimes with simpler and unified proofs) recent results on the structure of the set of nonnegative polynomials (the radius of the inscribed ball, volume estimates, and relations to the sums of squares). Our main tool is a new simple description of the ellipsoid of the largest volume contained in B ◦.
Faster Real Feasibility via Circuit Discriminants
, 2009
"... We show that detecting real roots for honestly nvariate (n + 2)nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. We then give a char ..."
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We show that detecting real roots for honestly nvariate (n + 2)nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. We then give a characterization of those functions k(n) such that the complexity of detecting real roots for nvariate (n + k(n))nomials transitions from P to NPhardness as n − → ∞. Our proofs follow in large part from a new complexity threshold for deciding the vanishing of Adiscriminants of nvariate (n+k(n))nomials. Diophantine approximation, through linear forms in logarithms, also arises as a key tool.
Volumes of nonnegative polynomials, sums of squares, and powers of linear forms
"... Abstract. We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We sh ..."
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Abstract. We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two it follows that there are significantly more nonnegative polynomials than sums of squares and there are significantly more sums of squares than sums of powers of linear forms. Moreover, we quantify the exact discrepancy between the cones; from our bounds it follows that the discrepancy grows as the number of variables increases. 1.
New Complexity Thresholds for Sparse Real Polynomials and Adiscriminants
, 2008
"... Let FEASR denote the problem of deciding whether a given system of real polynomial equations has a real root or not. While FEASR is arguably the most fundamental problem of real algebraic geometry, our current knowledge of its computational complexity is surprisingly coarse. This is a pity, for in a ..."
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Let FEASR denote the problem of deciding whether a given system of real polynomial equations has a real root or not. While FEASR is arguably the most fundamental problem of real algebraic geometry, our current knowledge of its computational complexity is surprisingly coarse. This is a pity, for in addition to numerous practical applications [BGV03], FEASR is also an important
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"... The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases ..."
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The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases