A convex geometric approach to counting the roots of a polynomial system

Rojas, J. Maurice

doi:10.1016/0304-3975(93)00062-a

Cited by 45 publications

(38 citation statements)

References 26 publications

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“…Our bound is significantly tighter than the root count suggested in [9]. A simple explanation derives from the monotonicity of the mixed volume: growing polytopes potentially increases (and never decreases) the mixed volume; and the support sets for which we compute the mixed volumes here are usually much smaller than the shadowed sets used in [9]. Furthermore, our bound appears to be easier to compute.…”

Section: ±1mentioning

confidence: 74%

“…While this bound is, in general, significantly sharper than the Bézout number (and its variants [11]), it only counts the zeros of P (x) in the n-dimensional algebraic torus (C * ) n . In [9], J.M. Rojas proposed a root count formula in C n , derived from the theory of toric varieties, and its computation involves finding the mixed volume of n shadowed polytopes in R n .…”

Section: ±1mentioning

confidence: 99%

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The BKK root count in $\mathbf {C}^n$

Wang

1996

Math. Comp.

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Abstract. The root count developed by Bernshtein, Kushnirenko and Khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus (C * ) n . In this paper, we modify this bound slightly so that it counts the number of isolated zeros in C n . Our bound is, apparently, significantly sharper than the recent root counts found by Rojas and in many cases easier to compute. As a consequence of our result, the Huber-Sturmfels homotopy for finding all the isolated zeros of a polynomial system in (C * ) n can be slightly modified to obtain all the isolated zeros in C n .

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Section: ±1mentioning

confidence: 74%

Section: ±1mentioning

confidence: 99%

The BKK root count in $\mathbf {C}^n$

Wang

1996

Math. Comp.

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“…. , a n such that the above bound is at least as sharp as the shadowed mixed volume bounds in [Roj94] (cf. Theorem 3).…”

Section: Introductionmentioning

confidence: 90%

“…. , n͖ case is just the BKK bound over a general algebraically closed field [Dan78,Roj94]. So in this case we obtain an upper bound on the number of isolated roots whose coordinate are all nonzero.…”

Section: Introductionmentioning

confidence: 97%

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Counting Affine Roots of Polynomial Systems via Pointed Newton Polytopes

Rojas

Wang

1996

Journal of Complexity

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We give a new upper bound on the number of isolated roots of a polynomial system. Unlike many previous bounds, our bound can also be restricted to different open subsets of affine space. Our methods give significantly sharper bounds than the classical Bé zout theorems and further generalize the mixed volume root counts discovered in the late 1970s. We also give a complete combinatorial classification of the subsets of coefficients whose genericity guarantees that our bound is actually an exact root count in affine space. Our results hold over any algebraically closed field.

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Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets

Sommese

Verschelde

2000

Journal of Complexity

109

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Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem. Academic PressKey Words: polynomial system; numerical homotopy continuation; components of solutions; numerical algebraic geometry; generic points; embedding. 4. A worked out example. 5. Bertini 's theorem and a local extension theorem. 6. Applications and computational experiences. 6.1. A planar four-bar mechanism. Contents6.2. Constructing Runge-Kutta formulas. 6.3. On Fourier transforms: the cyclic n-roots problem. 7. Conclusions and future directions.

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A convex geometric approach to counting the roots of a polynomial system

Cited by 45 publications

References 26 publications

The BKK root count in $\mathbf {C}^n$

The BKK root count in $\mathbf {C}^n$

Counting Affine Roots of Polynomial Systems via Pointed Newton Polytopes

Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets

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