45,803 research outputs found

    A Polyhedral Method to Compute All Affine Solution Sets of Sparse Polynomial Systems

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    To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible decomposition of a variety is typically understood in affine space, including also those components with zero coordinates. We present a polyhedral method to compute all affine solution sets of a polynomial system. The method enumerates all factors contributing to a generalized permanent. Toric solution sets are recovered as a special case of this enumeration. For sparse systems as adjacent 2-by-2 minors our methods scale much better than the techniques from numerical algebraic geometry

    Real root finding for equivariant semi-algebraic systems

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    Let RR be a real closed field. We consider basic semi-algebraic sets defined by nn-variate equations/inequalities of ss symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2d<n2d < n. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most 2d12d-1 distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by ss polynomials of degree dd in time (sn)O(d)(sn)^{O(d)}. This improves the state-of-the-art which is exponential in nn. When the variables x1,,xnx_1, \ldots, x_n are quantified and the coefficients of the input system depend on parameters y1,,yty_1, \ldots, y_t, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time (sn)O(dt)(sn)^{O(dt)}

    Transfer Function Synthesis without Quantifier Elimination

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    Traditionally, transfer functions have been designed manually for each operation in a program, instruction by instruction. In such a setting, a transfer function describes the semantics of a single instruction, detailing how a given abstract input state is mapped to an abstract output state. The net effect of a sequence of instructions, a basic block, can then be calculated by composing the transfer functions of the constituent instructions. However, precision can be improved by applying a single transfer function that captures the semantics of the block as a whole. Since blocks are program-dependent, this approach necessitates automation. There has thus been growing interest in computing transfer functions automatically, most notably using techniques based on quantifier elimination. Although conceptually elegant, quantifier elimination inevitably induces a computational bottleneck, which limits the applicability of these methods to small blocks. This paper contributes a method for calculating transfer functions that finesses quantifier elimination altogether, and can thus be seen as a response to this problem. The practicality of the method is demonstrated by generating transfer functions for input and output states that are described by linear template constraints, which include intervals and octagons.Comment: 37 pages, extended version of ESOP 2011 pape
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