On the intrinsic complexity of point finding in real singular hypersurfaces

In previous work we designed an efficient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of a non–smooth real hypersurface by means of a reduction to a smooth complete intersection

Nearest Points on Toric Varieties

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.Comment: 20 page

Polar Varieties, Real Equation Solving and Data-Structures: The hypersurface case

In this paper we apply for the first time a new method for multivariate equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for complex root determination to the {\em real} case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of \cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem--adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in non-uniform sequential time that is {\em polynomial} in the length of the input (given in straight--line program representation) and an adequately defined {\em geometric degree of the equation system}. Replacing the notion of geometric degree of the given polynomial equation system by a suitably defined {\em real (or complex) degree} of certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above.Comment: Late

Dualities in Convex Algebraic Geometry

Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre's representation as a spectrahedral shadow, and we explore the geometric underpinnings of semidefinite programming duality.Comment: 48 pages, 11 figure

Learning Algebraic Varieties from Samples

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package

Extremal and typical results in Real Algebraic Geometry

In the first part of the dissertation we show that $2((d-1)^n-1)/(d-2)$ is the maximum possible number of critical points that a generic $(n-1)$-dimensional spherical harmonic of degree $d$ can have. Our result in particular shows that there exist generic real symmetric tensors whose all eigenvectors are real. The results of this part are contained in Chapter \ref{ch:harmonics}. In the second part of the thesis we are interested in expected outcomes in three different problems of probabilistic real algebraic and differential geometry. First, in Chapter \ref{ch:discriminant} we compute the volume of the projective variety \Delta\subset \txt{P}\txt{Sym}(n,\K{R}) of real symmetric matrices with repeated eigenvalues. Our computation implies that the expected number of real symmetric matrices with repeated eigenvalues in a uniformly distributed projective $2$-plane L\subset \txt{P}\txt{Sym}(n,\K{R}) equals \mean\#(\Delta\cap L) = {n\choose 2}. The sharp upper bound on the number of matrices in the intersection $\Delta\cap L$ of $\Delta$ with a generic projective $2$-plane $L$ is ${n+1 \choose 3}$. Second, in Chapter \ref{ch:pevp} we provide explicit formulas for the expected condition number for the polynomial eigenvalue problem defined by matrices drawn from various Gaussian matrix ensembles. Finally, in Chapter \ref{ch:tangents} we are interested in the expected number of lines that are simultaneously tangent to the boundaries of several convex sets randomly positioned in the sphere. We express this number in terms of the integral mean curvatures of the boundaries of the convex sets