Abstract
Given a polynomial system f, this article provides a general construction for homotopies that yield at least one point of each connected component on the set of solutions of \(f = 0\). This algorithmic approach is then used to compute a superset of the isolated points in the image of an algebraic set which arises in many applications, such as computing critical sets used in the decomposition of real algebraic sets. An example is presented which demonstrates the efficiency of this approach.
D. J. Bates—Supported in part by AFOSR grant FA8650-13-1-7317, NSF ACI-1440467, and NSF DMS-1719658.
D. A. Brake and J. D. Hauenstein—Supported in part by AFOSR grant FA8650-13-1-7317 and NSF ACI-1460032.
A. J. Sommese—Supported in part by the Duncan Chair of the University of Notre Dame, AFOSR grant FA8650-13-1-7317, and NSF ACI-1440607.
C. W. Wampler—Supported in part by AFOSR grant FA8650-13-1-7317.
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Notes
- 1.
In usual practice, “randomization” means replacing a set of polynomials with some number of random linear combinations of the polynomials. When the appropriate number of combinations is used, then in a Zariski-open subset of the Cartesian space of coefficients of the linear combinations, the solution set of interest is preserved. See, for example, [25, Sect. 13.5]. Here, for simplicity of illustration, we take very simple linear combinations involving small integers. These happen to suffice, but in general one would use a random number generator and possibly hundreds of digits to better approximate the probability-one chance of success that is implied in a continuum model of the coefficient space.
- 2.
As before, we choose simple rational coefficients for simplicity of presentation.
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Bates, D.J., Brake, D.A., Hauenstein, J.D., Sommese, A.J., Wampler, C.W. (2017). Homotopies for Connected Components of Algebraic Sets with Application to Computing Critical Sets. In: Blömer, J., Kotsireas, I., Kutsia, T., Simos, D. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2017. Lecture Notes in Computer Science(), vol 10693. Springer, Cham. https://doi.org/10.1007/978-3-319-72453-9_8
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