Experiments on the Zeros of Harmonic Polynomials Using Certified Counting

REFERENCES

• [Abramowitz and Stegun 72] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions. Dover, 1972.
   Google Scholar
• [Bates et al. 06] D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler. “Software for Numerical Algebraic Geometry.” Available online (bertini.nd.edu), 2006.
   Google Scholar
• [Bates et al. 13] D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler. Numerically Solving Polynomial Systems with Bertini, Software, Environments, and Tools 25. SIAM, 2013.
   Google Scholar
• [Blum et al. 98] L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer, 1998.
   Google Scholar
• [Bshouty and Lyzzaik 10] D. Bshouty and A. Lyzzaik. “Problems and Conjectures for Planar Harmonic Mappings.” In Proceedings of the ICM2010 Satellite Conference: International Workshop on Harmonic and Quasiconformal Mappings (HQM2010), pp. 69–82. Special issue in J. Analysis 18 (2010).
   Google Scholar
• [Bshouty et al. 95] D. Bshouty, W. Hengartner, and T. Suez. “The Exact Bound on the Number of Zeros of Harmonic Polynomials.” J. Anal. Math. 67 (1995), 207–218.
   Web of Science ®Google Scholar
• [Edelman and Kostlan 95] A. Edelman and E. Kostlan. “How Many Zeros of a Random Polynomial Are Real?” Bull. Amer. Math. Soc. 32 (1995), 1–37.
   Web of Science ®Google Scholar
• [Geyer 08] L. Geyer. “Sharp Bounds for the Valence of Certain Harmonic Polynomials.” Proc. AMS 136 (2008), 549–555.
   Web of Science ®Google Scholar
• [Granville and Wigman 11] A. Granville and I. Wigman. “The Distribution of the Number of Zeros of Trigonometric Polynomials.” Amer. J. of Math. 133 (2011), 295–357.
   Web of Science ®Google Scholar
• [Greene et al. 13] B. Greene, D. Kagan, A. Masoumi, D. Mehta, E. J. Weinberg, and X. Xiao. “Tumbling through a Landscape: Evidence of Instabilities in High-Dimensional Moduli Spaces.” Phys. Rev. D 88 (2013), 026005.
   Web of Science ®Google Scholar
• [Hauenstein and Sottile 12] J. D. Hauenstein and F. Sottile. “Algorithm 921: AlphaCertified: Certifying Solutions to Polynomial Systems.” ACM TOMS 38 (2012), 28.
   Web of Science ®Google Scholar
• [He et al. 13] Y.-H. He, D. Mehta, M. Niemerg, M. Rummel, and A. Valeanu. “Exploring the Potential Energy Landscape over a Large Parameter-Space.” J. High Energy Phys. 1307 (2013), 050.
   Web of Science ®Google Scholar
• [Hughes et al. 13] C. Hughes, D. Mehta, and J.-I. Skullerud. “Enumerating Gribov Copies on the Lattice.” Annals Phys. 331 (2013), 188–215.
   Web of Science ®Google Scholar
• [Kastner and Mehta 11] M. Kastner and D. Mehta. “Phase Transitions Detached from Stationary Points of the Energy Landscape.” Phys. Rev. Lett. 107 (2011), 160602.
   PubMed Web of Science ®Google Scholar
• [Khavinson and Swiatek 03] D. Khavinson and G. Swiatek. “On a Maximal Number of Zeros of Certain Harmonic Polynomials.” Proc. AMS 131 (2003), 409–414.
   Web of Science ®Google Scholar
• [Lee et al. 15] S-Y. Lee, A. Lerario, and E. Lundberg. “Remarks on Wilmshurst’s Theorem.” To appear in Indiana U. Math. J., 2015.
   Google Scholar
• [Li and Wei 09] W. V. Li and A. Wei. “On the Expected Number of Zeros of Random Harmonic Polynomials.” Proc. AMS 137 (2009), 195–204.
   Web of Science ®Google Scholar
• [Maniatis and Mehta 12] M. Maniatis and D. Mehta. “Minimizing Higgs Potentials via Numerical Polynomial Homotopy Continuation.” Eur. Phys. J. Plus 127 (2012), 91.
   Web of Science ®Google Scholar
• [Martinez-Pedrera et al. 13] D. Martinez-Pedrera, D. Mehta, M. Rummel, and A. Westphal. “Finding All Flux Vacua in an Explicit Example.” J. High Energy Phys. 1306 (2013), 110.
   Web of Science ®Google Scholar
• [Mehta 09] D. Mehta. “Lattice vs. Continuum: Landau Gauge Fixing and ’t Hooft-Polyakov Monopoles.” PhD thesis, University of Adelaide, Australasian Digital Theses Program, 2009.
   Google Scholar
• [Mehta 11a] D. Mehta. “Finding All the Stationary Points of a Potential Energy Landscape via Numerical Polynomial Homotopy Continuation Method.” Phys. Rev. E 84 (2011), 025702.
   Web of Science ®Google Scholar
• [Mehta 11b] D. Mehta. “Numerical Polynomial Homotopy Continuation Method and String Vacua.” Adv. High Energy Phys. 2011 (2011), 263937.
   Web of Science ®Google Scholar
• [Mehta et al. 09] D. Mehta, A. Sternbeck, L. von Smekal, and A. G. Williams. “Lattice Landau Gauge and Algebraic Geometry.” Proceedings of Science QCD-TNT09 (2009), 025.
   Google Scholar
• [Mehta et al. 12a] D. Mehta, Y.-H. He, and J. D. Hauenstein. “Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories.” J. High Energy Phys. 1207 (2012), 018.
   Web of Science ®Google Scholar
• [Mehta et al. 12b] D. Mehta, J. D. Hauenstein, and M. Kastner. “Energy Landscape Analysis of the Two-Dimensional Nearest-Neighbor φ4 Model.” Phys. Rev. E 85 (2012), 061103.
   Web of Science ®Google Scholar
• [Mehta et al. 13a] D. Mehta, D. A. Stariolo, and M. Kastner. “Energy Landscape of the Finite-Size Spherical Three-Spin Glass Model.” Phys. Rev. E 87 (2013), 052143.
   Web of Science ®Google Scholar
• [Mehta et al. 13b] D. Mehta, J. D. Hauenstein, and D. J. Wales. “Certifying the Potential Energy Landscape.” J. Chem. Phys. 138 (2013), 171101.
   PubMed Web of Science ®Google Scholar
• [Peretz and Schmid 97] R. Peretz and J. Schmid. “On the Zero Sets Of Certain Complex Polynomials.” In Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), Israel Math. Conf. Proc. 11, pp. 203–208. Bar-Ilan Univ. Ramat Gan, 1997.
   Google Scholar
• [Sheil-Small 02] T. Sheil-Small. Complex Polynomials. Cambridge University Press, 2002.
   Google Scholar
• [Sommese and Wampler 05] A. J. Sommese and C. W. Wampler. The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific, 2005.
   Google Scholar
• [Wilmshurst 98] A. S. Wilmshurst. “The Valence of Harmonic Polynomials.” Proc. AMS 126 (1998), 2077–2081.
   Web of Science ®Google Scholar