Batkhin, A. B.; Bruno, A. D. Investigation of a real algebraic surface. (English. Russian original) Zbl 1349.14178 Program. Comput. Softw. 41, No. 2, 74-83 (2015); translation from Programmirovanie 41, No. 2 (2015). Summary: A description of a real algebraic variety in \(\mathbb{R}^{3}\) is given. This variety plays an important role in the investigation of the Einstein metrics whose evolution is studied using the normalized Ricci flow. To reveal the internal structure of this variety, a description of all its singular points is given. Due to the internal symmetry of this variety, a part of the investigation uses elementary symmetric polynomials. All the computations are performed using computer algebra algorithms (in particular, Gröbner bases) and algorithms for dealing with polynomial ideals. As an auxiliary result, a proposition about the structure of the discriminant surface of a cubic polynomial is proved. Cited in 4 Documents MSC: 14P05 Real algebraic sets 14Q10 Computational aspects of algebraic surfaces 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 53A05 Surfaces in Euclidean and related spaces 68W30 Symbolic computation and algebraic computation Software:desing; PGeomlib PDF BibTeX XML Cite \textit{A. B. Batkhin} and \textit{A. D. Bruno}, Program. Comput. Softw. 41, No. 2, 74--83 (2015; Zbl 1349.14178); translation from Programmirovanie 41, No. 2 (2015) Full Text: DOI References: [1] Abiev, N.A., Arvanitoyeorgos, A., Nikonorov, Yu.G., and Siasos, P., The dynamics of the Ricci flow on generalized Wallach spaces, Differ. Geom. Appl., 2014, vol. 35, pp. 26-43. · Zbl 1327.53062 [2] Abiev, NA; Arvanitoyeorgos, A.; Nikonorov, YuG; Siasos, P.; Rovenski, V. (ed.); Walczak, P. (ed.), The Ricci flow on some generalized Wallach spaces, No. 72, 3-37 (2014) · Zbl 1323.53069 [3] Abiev, NA; Arvanitoyeorgos, A.; Nikonorov, YuG; Siasos, P., The normalized Ricci flow on generalized Wallach spaces, Mat. Forum, vol. 8 (1), 25-42 (2014), Vladikavkaz [4] Finikov, S.P., Theory of Surfaces, Moscow: GTTI, 1934 [in Russian]. [5] Cox, D., Little, J., and O’Shea, D., Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, New York: Springer, 2007. · Zbl 1118.13001 [6] Bruno, A.D. and Batkhin, A.B., Resolution of an algebraic singularity by power geometry algorithms, Program. Comput. Software, 2012, vol. 38, no. 2, pp. 57-72. · Zbl 1253.13022 [7] Batkhin, A.B., Bruno, A.D., and Varin, V. P., Stability sets of multiparameter Hamiltonian systems, J. Appl. Math. Mech., 2012, vol. 76, no. 1, pp. 56-92. · Zbl 1272.70076 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.