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Smale's alpha-theory uses estimates related to the convergence of Newton's method to give criteria implying that Newton iterations will converge quadratically to solutions to a square polynomial system.
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by same author, Stepsize control for path tracking , Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 21–31
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J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry , C. R. Math. Acad. Sci. Paris 336
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L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann, MPFR: a multiple-precision binary floating-point library with correct rounding , ACM Trans. Math. Software 33
2007
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D.J. Bates, J.D. Hauenstein, A.J. Sommese, and C.W. Wampler, II, Adaptive multiprecision path tracking , SIAM J. Numer. Anal. 46
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D.J. Bates, J.D. Hauenstein, C. Peterson, and A.J. Sommese, Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials , Approximate commutative algebra, Texts Monogr. Symbol. Comput., Springer, Vienna, 2009, pp. 55–77
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E. Mukhin, V. Tarasov, and A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz , Ann. of Math. (2) 170
2009
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D.J. Bates, J.D. Hauenstein, and A.J. Sommese, Efficient path tracking methods , Preprint, 2010
2010
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W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular
2010
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L. García-Puente, N. Hein, C. Hillar, A. Martín del Campo-Sanchez, J. Ruffo, F. Sottile, and Z. Teitler, The Secant Conjecture in the real Schubert calculus , arXiv.org/1010.0665
2010
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G. Jeronimo and D. Perrucci, On the minimum of a positive polynomial over the standard simplex , J. Symbolic Comput. 45
2010
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F. Sottile, Frontiers of reality in Schubert calculus , Bull. Amer. Math. Soc. (N.S.) 47
2010
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J.D. Hauenstein, A.J. Sommese, and C.W. Wampler, Regeneration homotopies for solving systems of polynomials , Math. Comp. 80
2011
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