Faugère, Jean-Charles; Hering, Milena; Phan, Jeff The membrane inclusions curvature equations. (English) Zbl 1065.92005 Adv. Appl. Math. 31, No. 4, 643-658 (2003). Summary: We examine a system of equations arising in biophysics whose solutions are believed to represent the stable positions of \(N\) conical proteins embedded in a cell membrane. Symmetry considerations motivate two equivalent reformulations of the system which allow the complete classification of solutions for small \(N<13\). The occurrence of regular geometric patterns in these solutions suggests considering a simpler system, which leads to the detection of solutions for larger \(N\) up to 280. We use the most recent techniques of Gröbner bases computation for solving polynomial systems of equations. Cited in 1 Document MSC: 92C05 Biophysics 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 92C40 Biochemistry, molecular biology Software:Maple; SF; FGb PDFBibTeX XMLCite \textit{J.-C. Faugère} et al., Adv. Appl. Math. 31, No. 4, 643--658 (2003; Zbl 1065.92005) Full Text: DOI References: [1] A. Lascoux, S. Veigneau, Algebraic combinatorics environment, Technical report, Institut Gaspard Monge, Université de Marne-la-Vallée, 1998, Version 3.0; A. Lascoux, S. Veigneau, Algebraic combinatorics environment, Technical report, Institut Gaspard Monge, Université de Marne-la-Vallée, 1998, Version 3.0 [2] Becker, T.; Weispfenning, V., Gröbner bases, a computational approach to commutative algebra, Graduate Texts in Math. (1993), Springer-Verlag · Zbl 0772.13010 [3] B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassen-ringes nach einem nulldimensionalen Polynomideal, PhD thesis, Innsbruck, 1965; B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassen-ringes nach einem nulldimensionalen Polynomideal, PhD thesis, Innsbruck, 1965 · Zbl 1245.13020 [4] Buchberger, B., An algorithmical criterion for the solvability of algebraic systems, Aequationes Math., 4, 3, 374-383 (1970), in German [5] Buchberger, B., A criterion for detecting unnecessary reductions in the construction of Gröbner basis, (Proc. EUROSAM 79. Proc. EUROSAM 79, Lecture Notes in Comput. Sci., 72 (1979), Springer-Verlag), 3-21 · Zbl 0417.68029 [6] Buchberger, B., Gröbner bases: an algorithmic method in polynomial ideal theory, (Reidel, Recent Trends in Multidimensional System Theory (1985), Bose) · Zbl 0587.13009 [7] Char, B.; Geddes, K.; Gonnet, G.; Leong, B.; Monagan, M.; Watt, S., Maple V Library Reference Manual (1991), Springer-Verlag, third printing, 1993 · Zbl 0763.68046 [8] Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties and Algorithms (1992), Springer-Verlag: Springer-Verlag New York [9] Cox, D.; Little, J.; O’Shea, D., Using Algebraic Geometry (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0920.13026 [10] Davenport, J. H.; Siret, Y.; Tournier, E., Calcul Fortnel (1993), Masson [11] H. Derksen, Private communication, Seminar Berkeley, 1999; H. Derksen, Private communication, Seminar Berkeley, 1999 [12] D. Bini, G. Fiorentino, G. Mpsolve, Technical report, University of Pisa, 1999; D. Bini, G. Fiorentino, G. Mpsolve, Technical report, University of Pisa, 1999 [13] J.C. Faugère, FGb available online, available on the WEB http://calfor.lip6.fr/ jcf/FGb.html; J.C. Faugère, FGb available online, available on the WEB http://calfor.lip6.fr/ jcf/FGb.html [14] Faugère, J. C.; Gianni, P.; Lazard, D.; Mora, T., Efficient computation of zero-dimensional Gröbner basis by change of ordering, J. Symbolic Comput., 16, 4, 329-344 (1993) · Zbl 0805.13007 [15] J.C. Faugère, Résolution des systèmes d’équations algebriques, PhD thesis, Université Paris 6, février, 1994; J.C. Faugère, Résolution des systèmes d’équations algebriques, PhD thesis, Université Paris 6, février, 1994 [16] Faugère, J. C., A new efficient algorithm for computing Gröbner bases (F4), J. Pure Appl. Algebra, 139, 1-3, 61-88 (1999) · Zbl 0930.68174 [17] G.-M. Greuel, G. Pfister, H. Schoenemann, SINGULAR 1.2.3, February 1999, http://www.mathematik.uni-kl.de/ zca/Singular/Welcome.html; G.-M. Greuel, G. Pfister, H. Schoenemann, SINGULAR 1.2.3, February 1999, http://www.mathematik.uni-kl.de/ zca/Singular/Welcome.html [18] J. Stembridge, The symmetric functions package, Technical report, Maple share library, 1998; J. Stembridge, The symmetric functions package, Technical report, Maple share library, 1998 [19] K.S. Kim, Private communication, Cambridge, 1999; K.S. Kim, Private communication, Cambridge, 1999 [20] Kim, K. S.; Neu, J.; Oster, G., Curvature-mediated interactions between membrane proteins, Biophys. J., 15, 2214-2291 (1998) [21] Symmetrica Available on the web at http://www.mathe2.uni-bayreuth.de/axel/symneu_engl.html; Symmetrica Available on the web at http://www.mathe2.uni-bayreuth.de/axel/symneu_engl.html This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.