Abstract
Let \(R = \mathbb{C}[x_{1},\ldots,x_{N}]\) and let \(F =\{ f_{1},\ldots,f_{t}\} \subset R\) be a set of generators for an ideal I. Let \(Y =\{ y_{1},\ldots,y_{\ell}\} \subset {\mathbb{C}}^{N}\) be a subset of the set of isolated solutions of the zero locus of F. Let \(\mathfrak{m}_{y_{i}}\) denote the maximal ideal of y i and let \(\mathcal{P}_{y_{i}}\) denote the \(\mathfrak{m}_{y_{i}}\)-primary component of I. Let \(J = \cap _{i=1}^{l}\mathcal{P}_{y_{i}}\) and let \(\mathcal{Z}\) denote the corresponding zero dimensional subscheme supported on Y. This article presents a numerical algorithm for computing the Hilbert function and the regularity of \(\mathcal{Z}\). In addition, the algorithm produces a monomial basis for R∕J. The input for the algorithm is the polynomial system F and a numerical approximation of each element in Y.
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Bates, D.J., Hauenstein, J.D., Peterson, C., Sommese, A.J.: A numerical local dimension test for points on the solution set of a system of polynomial equations. SIAM J. Numer. Anal. 47(5), 3608–3623 (2009)
Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: software for numerical algebraic geometry. Available at bertini.nd.edu (2006)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (1996)
Dayton, B.H., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: ISSAC’05, pp. 116–123. ACM, New York (2005)
Hao, W., Sommese, A.J. and Zeng, Z.: Algorithm 931: An algorithm and software for computing multiplicity structures at zeros of nonlinear systems, Transactions of Mathematical Software, 40 (2013), Article 5: appears online at dx.doi.org/10.1145/2513109.2513114
Hauenstein, J.D., Sottile, F.: Algorithm 921: alphaCertified: certifying solutions to polynomial systems. Trans. Math. Softw. 38(4), 28 (2012)
Hauenstein, J.D., Wampler, C.W.: Isosingular sets and deflation. Found. Comput. Math. (2013). doi:10.1007/s10208-013-9147-y
Leykin, A., Verschelde, J., Zhao, A.: Newton’s method with deflation for isolated singularities of polynomial systems. Theor. Comp. Sci. 359, 111–122 (2006)
Leykin, A., Verschelde, J., Zhao, A.: Higher-order deflation for polynomial systems with isolated singular solutions. In: Algorithms in Algebraic Geometry. The IMA Volumes in Mathematics and Its Applications, vol. 146, pp. 79–97. Springer, New York (2008)
Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge University Press, Cambridge (1916)
Morgan, A.P., Sommese, A.J., Wampler, C.W.: Complete solution of the nine-point path synthesis problem for four-bar linkages. ASME J. Mech. Des. 114(1), 153–159 (1992)
Ojika, T.: Modified deflation algorithm for the solution of singular problems. J. Math. Anal. Appl. 123, 199–221 (1987)
Zeng, Z.: The closedness subspace method for computing the multiplicity structure of a polynomial system. Contemp. Math. 496, 347–362 (2009)
Acknowledgments
The second author was partially supported by NSF grant DMS-1262428. The third author was partially supported by NSF grant DMS-1228308. The fourth was partially supported by the Duncan Chair of the University of Notre Dame and DARPA/AFRL G-2457-2.
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Griffin, Z.A., Hauenstein, J.D., Peterson, C., Sommese, A.J. (2014). Numerical Computation of the Hilbert Function and Regularity of a Zero Dimensional Scheme. In: Cooper, S., Sather-Wagstaff, S. (eds) Connections Between Algebra, Combinatorics, and Geometry. Springer Proceedings in Mathematics & Statistics, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0626-0_6
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DOI: https://doi.org/10.1007/978-1-4939-0626-0_6
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