9 research outputs found

    Classification of triples of lattice polytopes with a given mixed volume

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    We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume mm in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed mm. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals

    Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes

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    Let P1,..., Pn and Q1,...,Qn be convex polytopes in Rn such that Pi is a proper subset of Qi . It is well-known that the mixed volume has the monotonicity property: V (P1,...,Pn) is less than or equal to V (Q1,...,Qn) . We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P1,..., Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 U...U Pn . In addition, we obtain an analog of Cramer\u27s rule for sparse polynomial systems

    Overdetermined Systems of Equations on Toric, Spherical, and Other Algebraic Varieties

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    Let E1,,EkE_1,\ldots,E_k be a collection of linear series on an algebraic variety XX over C\mathbb{C}. That is, EiH0(X,Li)E_i\subset H^0(X, \mathcal{L}_i) is a finite dimensional subspace of the space of regular sections of line bundles Li \mathcal{L}_i. Such a collection is called overdetermined if the generic system s1==sk=0, s_1 = \ldots = s_k = 0, with siEis_i\in E_i does not have any roots on XX. In this paper we study solvable systems which are given by an overdetermined collection of linear series. Generalizing the notion of a resultant hypersurface we define a consistency variety Ri=1kEiR\subset \prod_{i=1}^k E_i as the closure of the set of all systems which have at least one common root and study general properties of zero sets ZsZ_{\bf s} of a generic consistent system sR{\bf s}\in R. Then, in the case of equivariant linear series on spherical homogeneous spaces we provide a strategy for computing discrete invariants of such generic non-empty set ZsZ_{\bf s}. For equivariant linear series on the torus (C)n(\mathbb{C}^*)^n this strategy provides explicit calculations and generalizes the theory of Newton polyhedra.Comment: Improved exposition, minor changes. 18 pages, comments are welcome

    Inequalities between mixed volumes of convex bodies: volume bounds for the Minkowski sum

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    In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum P1++PdP_1+\dots+P_d of dd-dimensional lattice polytopes is bounded from above by a function of order O(m2d)O(m^{2^d}), where mm is the mixed volume of the tuple (P1,,Pd)(P_1,\dots,P_d). This is a consequence of the well-known Aleksandrov-Fenchel inequality. Esterov also posed the problem of determining a sharper bound. We show how additional relations between mixed volumes can be employed to improve the bound to O(md)O(m^d), which is asymptotically sharp. We furthermore prove a sharp exact upper bound in dimensions 2 and 3. Our results generalize to tuples of arbitrary convex bodies with volume at least one.Comment: 21 pages, 5 figures; error in the statement of Theorem 3.3 correcte

    Improvements to quantum search, with applications to cryptanalysis

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