Deflation Conjecture and Local Dimensions of Brent Equations

References

• Bates, D. J., Hauenstein, J. D., Peterson, C., Sommese, A. J. (2009). A numerical local dimensions test for points on the solution set of a system of polynomial equations. SIAM J. Numer. Anal. 47(5): 3608–3623. 10.1137/08073264X
   Web of Science ®Google Scholar
• Bates, D. J., Hauenstein, J. D., Peterson, C., Sommese, A. J. (2010). Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials. In: Robbiano, L., Abbott, J., eds. Approximate Commutative Algebra. Vienna: Springer, pp. 55–77.
   Google Scholar
• Bates, D. J., Hauenstein, J. D., Sommese, A. J., Wampler, C. W. (2013). Numerically Solving Polynomial Systems with Bertini. Software, Environments, and Tools, Vol. 25. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
   Google Scholar
• Bates, D. J., Eklund, D., Hauenstein, J. D., Peterson, C. (2021). Excess intersections and numerical irreducible decompositions. In: 2021 23rd International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), pp. 52–60.
   Google Scholar
• Burichenko, V. P. (2015). Symmetries of matrix multiplication algorithms I. arXiv:1508.01110.
   Google Scholar
• Cox, D. A., Little, J., O’Shea, D. (2015). Ideals, Varieties, and Algorithms. 4th ed. Undergraduate Texts in Mathematics. Cham: Springer.
   Google Scholar
• Dayton, B. H., Zeng, Z. (2005). Computing the multiplicity structure in solving polynomial systems. In: ISSAC’05. New York: ACM, pp. 116–123. 10.1145/1073884.1073902
   Google Scholar
• Dayton, B. H., Li, T.-Y., Zeng, Z. (2011). Multiple zeros of nonlinear systems. Math. Comp. 80(276): 2143–2168. 10.1090/S0025-5718-2011-02462-2
   Web of Science ®Google Scholar
• de Groote, H. F. (1978). On varieties of optimal algorithms for the computation of bilinear mappings. I. The isotropy group of a bilinear mapping. Theor. Comput. Sci. 7: 1–24. 10.1016/0304-3975(78)90038-5
   Google Scholar
• de Groote, H. F. (1978). On varieties of optimal algorithms for the computation of bilinear mappings. II. Optimal algorithms for 2 × 2 matrix multiplication. Theor. Comput. Sci. 7: 127–148. 10.1016/0304-3975(78)90045-2
   Google Scholar
• Decker, W., Schreyer, F. O. (2007). Varieties, Gröbner bases, and algebraic curves, to appear.
   Google Scholar
• de Silva, V., Lim, L.-H. (2008). Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3): 1084–1127. 10.1137/06066518X
   Web of Science ®Google Scholar
• Fawzi, A., et al. (2022). Discovering faster matrix multiplication algorithms with reinforcement learning. Nature 610: 47–53. 10.1038/s41586-022-05172-4
   PubMed Web of Science ®Google Scholar
• Fawzi, A., et al. (2022). Discovering faster matrix multiplication algorithms with reinforcement learning. Github: https://github.com/deepmind/alphatensor/.
   Google Scholar
• Harris, K., Hauenstein, J. D., Szanto, A. (2023). Smooth points on semi-algebraic sets. J. Symbolic Comput. 116: 183–212. 10.1016/j.jsc.2022.09.003
   Web of Science ®Google Scholar
• Hauenstein, J. D., Wampler, C. W. (2013). Isosingular sets and deflation. Found. Comput. Math. 13(3): 371–403. 10.1007/s10208-013-9147-y
   Web of Science ®Google Scholar
• Heule, M. J. H., Kauers, M., Seidl, m. Matrix multiplication repository. http://www.algebra.uni-linz.ac.at/research/matrix-multiplication/. Accessed: 2019-04-26.
   Google Scholar
• Heule, M. J. H., Kauers, M., Seidl, M. (2021). New ways to multiply 3 × 3-matrices. J. Symbolic Comput. 104: 899–916. 10.1016/j.jsc.2020.10.003
   Web of Science ®Google Scholar
• Hillar, C. J., Lim, L.-H. (2013). Most tensor problems are NP-hard. J. ACM 60(6): Art.45, 39pp. 10.1145/2512329
   Web of Science ®Google Scholar
• Kauers, M., Moosbauer, J. (2023). Flip graphs for matrix multiplication. In: ISSAC’23. Tromso, Norway: ACM, pp. 381–388. 10.1145/3597066.3597120
   Google Scholar
• Kauers, M., Moosbauer, J. (2023). Some new non-commutative matrix multiplication algorithms of size (n,m,6) . arXiv:2306.00882v1. https://github.com/jakobmoosbauer/flips
   Google Scholar
• Kuo, Y. C., Li, T. Y. (2008). Determining dimension of the solution component that contains a computed zero of a polynomial system. J. Math. Anal. Appl. 338(2): 840–851. 10.1016/j.jmaa.2007.05.049
   Web of Science ®Google Scholar
• Laderman, J. D. (1976). A noncommutative algorithm for multiplying 3 × 3 matrices using 23 multiplications. Bull. Amer. Math. Soc. 82(1): 126–128. 10.1090/S0002-9904-1976-13988-2
   Web of Science ®Google Scholar
• Landsberg, J. M. (2017). Geometry and Complexity Theory. Cambridge Studies in Advanced Mathematics, 169. Cambridge: Cambridge University Press.
   Google Scholar
• Lairez, P., Safey El Din, M. (2021). Computing the dimension of real algebraic sets. In: ISSAC’21, New York: ACM, pp. 257–264. 10.1145/3452143.3465551
   Google Scholar
• Leykin, A., Verschelde, J., Zhao, A. (2006). Newton’s method with deflation for isolated singularities of polynomial systems. Theoret. Comput. Sci. 359(1–3): 111–122. 10.1016/j.tcs.2006.02.018
   Web of Science ®Google Scholar
• Lewis, A. D. (2009). Semicontinuity of rank and nullity and some consequences. https://mast.queensu.ca/∼andrew/notes/pdf/2009a.pdf, preprint.
   Google Scholar
• Li, X., Bao, Y., Zhang, L. On the local dimensions of solutions of Brent equations, arXiv:2303.09754v2.
   Google Scholar
• Li, X. Github: https://github.com/Xinli-zjut.
   Google Scholar
• Ojika, T., Watanabe, S., Mitsui, T. (1983). Deflation algorithm for the multiple roots of a system of nonlinear equations. J. Math. Anal. Appl. 96(2): 463–479. 10.1016/0022-247X(83)90055-0
   Web of Science ®Google Scholar
• Ojika, T. (1987). Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations. J. Math. Anal. Appl. 123(1): 199–221. 10.1016/0022-247X(87)90304-0
   Web of Science ®Google Scholar
• Sommese, A. J., Wampler, C. W. (2005). The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. Singapore: World Scientific.
   Google Scholar
• Smirnov, A. V. (2013). The bilinear complexity and practical algorithms for matrix multiplication. Comput. Math. Math. Phys. 53(12): 1781–1795. 10.1134/S0965542513120129
   Web of Science ®Google Scholar
• Strassen, V. (1969). Gaussian elimination is not optimal. Numer. Math. 13: 354–356. 10.1007/BF02165411
   Web of Science ®Google Scholar
• Tichavský, P. (2021). Characterization of decomposition of matrix multiplication tensors. arXiv:2104.05323v1.
   Google Scholar
• Wampler, C. W., Hauenstein, J. D., Sommese, A. J. (2011). Mechanism mobility and a local dimension test. Mech. Mach. Theory 46: 1193–1206. 10.1016/j.mechmachtheory.2011.04.011
   Web of Science ®Google Scholar
• Zeng, Z. (2023). A Newton’s iteration converges quadratically to nonisolated solutions too. Math. Comp. 92(344): 2795–2824. 10.1090/mcom/3657
   Web of Science ®Google Scholar