# Read e-book Polynomial approximation on polytopes

### Kundrecensioner

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Please note that corrections may take a couple of weeks to filter through the various RePEc services. Economic literature: papers , articles , software , chapters , books. Random walk in a simplex and quadratic optimization over convex polytopes. In this paper we develop probabilistic arguments for justifying thequality of an approximate solution for global quadratic minimization problem, obtained as a best point among all points of a uniform grid inside a polyhedral feasible set. In this context, it makes sense to study any cones that can be optimized over efficiently.

## Polynomial Approximation on Polytopes - Vilmos Totik - Häftad () | Bokus

A prominent example is the positive semi-definite cone. Let us define as the set of real, symmetric matrices with non-negative eigenvalues. A spectrahedron is the intersection with an affine subspace. The value is referred to as the dimension of the spectrahedron.

In analogy with the parameter we defined for polyhedral lifts, let us define for a polytope to be the minimal dimension of a spectrahedron that linearly projects to. It is an exercise to show that for every polytope. In other words, spectahedral lifts are at least as powerful as polyhedral lifts in this model. In fact, they are strictly more powerful. Certainly there are many examples of this in the setting of approximation like the Goemans-Williamson SDP mentioned earlier , but there are also recent gaps between and for polytopes; see the work of Fawzi, Saunderson, and Parrilo.

## Convex volume approximation

Nevertheless, we are recently capable of proving strong lower bounds on the dimension of such lifts. Let us consider the cut polytope as in previous posts. Theorem 1 L-Raghavendra-Steurer There is a constant such that for every , one has. Our goal in this post and the next is to explain the proof of this theorem and how quantum entropy maximization plays a key role. Let be a non-negative matrix. One defines the psd rank of as the quantity. Theorem 2 For every polytope , it holds that for any slack matrix of. Recall the class of non-negative quadratic multi-linear functions that are positive on and the matrix given by.

We saw previously that is a submatrix of some slack matrix of. Thus our goal is to prove a lower bound on.

## Department of Mathematics

But now our proof system is considerably more powerful. For a subspace of functions , let us define the cone. This is the cone of squares of functions in. We will think of as a set of axioms of size that is able to assert non-negativity of every by writing. Fix a subspace and let. Fix also a basis for. Define by setting.

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- Polynomial Approximation on Polytopes.
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Note that is PSD for every because where. We can write every as. Defining by , we see that. As a difference, our method employs a tropical convex hull computation, and it exploits a particular kind of duality which is special for the tropical cones arising. This duality can be seen as a generalization of the Alexander duality of monomial ideals. Special families of these polynomials have appeared in economics, discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to many open problems of interest in discrete geometry.

This is a joint work with Ngoc Mai Tran. We slightly extend results by Sturmfels, Tevelev and Yu, but our point of view can be generalized to deal with non-generic parametrizations. In the case of a generic rational hypersurface S, we give combinatorial conditions on the relative positions of the Newton polytopes of the polynomials defining the parametrization to determine the degree and the order of singularity of S at the origin.

### some mathematics & computation

Some of these conditions look similar to those presented recently by Bihan and Soprunov, but they are different. In this talk, we focus on the interior of the maximal cones in the moduli space and classify all curves which can be realized as a faithful tropicalization in a tropical plane.

Reflecting the algebro-geometric world, we show that these are exactly those which are not realizably hyperelliptic. Our approach is constructive: For any not realizably hyperelliptic curve, we explicitly construct a realizable model of the tropical plane and a faithfully tropicalized quartic in it. These constructions rely on modifications resp.

Conversely, we prove that any realizably hyperelliptic curve cannot be embedded in such a fashion. For that, we rely on the theory of tropical divisors and embeddings from linear systems, and recent advances in the realizability of sections of the tropical canonical divisor. In this talk we will see how that allows us to compute the moduli space of Harnack curves with a given Newton polygon. We will show how this space can be embedded into the moduli space of pointed tropical curves.

Through this embedding the moduli space of Harnack curves has a compactification which essentially is the secondary polytope of the Newton polygon. This approach generalizes the polyhedral homotopies by Huber and Sturmfels. Joint work with Josephine Yu. The Gauss map represents a classical and powerful tool in projective geometry. I will define higher order Gauss mapps capturing higher osculating properties of the embedding and show that similar finiteness properties hold.

The talk is based on joint work with K.