# American Institute of Mathematical Sciences

September  2014, 6(3): 319-333. doi: 10.3934/jgm.2014.6.319

## Discriminantly separable polynomials and quad-equations

 1 The Department of Mathematical Sciences, University of Texas at Dallas, 800 West Campbell Road, Richardson TX 75080, United States 2 Faculty for Traffic and Transport Engineering, University of Belgrade, Vojvode Stepe 305, 11000 Belgrade, Serbia

Received  April 2013 Revised  July 2014 Published  September 2014

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable each. Our classification is based on the study of structures of zeros of a polynomial component $P$ of a discriminant. This classification is related to the classification of pencils of conics in a delicate way. We establish a relationship between our classification and the classification of integrable quad-equations which has been suggested recently by Adler, Bobenko, and Suris.
Citation: Vladimir Dragović, Katarina Kukić. Discriminantly separable polynomials and quad-equations. Journal of Geometric Mechanics, 2014, 6 (3) : 319-333. doi: 10.3934/jgm.2014.6.319
##### References:
 [1] V. E. Adler, A. I. Bobenko and Y. B. Suris, Classification of integrable equations on quad-graphs. The consistency approach,, Commun. Math. Phys., 233 (2003), 513. [2] V. E. Adler, A. I. Bobenko and Y. B. Suris, Discrete nonlinear hiperbolic equations. Classification of integrable cases,, Funct. Anal. Appl, 43 (2009), 3. doi: 10.1007/s10688-009-0002-5. [3] V. E. Adler, A. I. Bobenko and Yu. B. Suris, Geometry of Yang-Baxter maps: Pencils of conics and quadrirational mappings,, Comm. Anal. Geom., 12 (2004), 967. doi: 10.4310/CAG.2004.v12.n5.a1. [4] A. I. Bobenko and Yu. B. Suris, Integrable noncommutative equations on quad-graphs. The consistency approach,, Lett. Math. Phys., 61 (2002), 241. doi: 10.1023/A:1021249131979. [5] A. I. Bobenko and Yu. B. Suris, Integrable systems on quad-graphs,, Int. Math. Res. Not., (2002), 573. doi: 10.1155/S1073792802110075. [6] V. Buchstaber, n-valued groups: Theory and applications,, Moscow Mathematical Journal, 6 (2006), 57. [7] V. M. Buchstaber, Functional equations, associated with addition theorems for elliptic functions, and two-valued algebraic groups,, Russian Math. Surv., 45 (1990), 213. doi: 10.1070/RM1990v045n03ABEH002361. [8] V. M. Buchstaber and V. Dragović, Two-valued groups, Kummer varieties and integrable billiards,, preprint, (). [9] V. M. Buchstaber and S. P. Novikov, Formal groups, power systems and Adams operators,, Mat. Sb. (N. S), 84(126) (1971), 81. [10] V. M. Buchstaber and A. P. Veselov, Integrable correspondences and algebraic representations of multivalued groups,, Internat. Math. Res. Notices, 8 (1996), 381. doi: 10.1155/S1073792896000256. [11] G. Darboux, Principes de Géométrie Analytique,, Gauthier-Villars, (1917). [12] V. Dragović, Poncelet-Darboux curves, their complete decomposition and Marden theorem,, Int. Math. Res. Notes, 2011 (2011), 3502. doi: 10.1093/imrn/rnq229. [13] V. Dragović, Generalization and geometrization of the Kowalevski top,, Communications in Math. Phys., 298 (2010), 37. doi: 10.1007/s00220-010-1066-z. [14] V. Dragović and K. Kukić, New examples of systems of the Kowalevski type,, Regular and Chaotic Dynamics, 16 (2011), 484. doi: 10.1134/S1560354711050054. [15] V. Dragović and K. Kukić, Systems of the Kowalevski type and discriminantly separable polynomials,, Regular and Chaotic Dynamics, 19 (2014), 162. doi: 10.1134/S1560354714020026. [16] V. Dragović and M. Radnović, Poncelet Porisms and Beyond,, Springer, (2011). doi: 10.1007/978-3-0348-0015-0. [17] V. Dragović and M. Radnović, Billiard algebra, integrable line congruences and DR-nets,, J. Nonlinear Mathematical Physics, 19 (2012). doi: 10.1142/S1402925112500192. [18] V. V. Golubev, Lectures on the Integration of Motion of a Heavy Rigid Body Around a Fixed Point,, Gostechizdat, (1953). [19] S. Kowalevski, Sur la probleme de la rotation d'un corps solide autour d'un point fixe,, Acta Math., 12 (1889), 177. doi: 10.1007/BF02592182. [20] J. G. Semple and G. T. Kneebone, Algebraic Projective Geometry,, Clarendon Press, (1998).

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##### References:
 [1] V. E. Adler, A. I. Bobenko and Y. B. Suris, Classification of integrable equations on quad-graphs. The consistency approach,, Commun. Math. Phys., 233 (2003), 513. [2] V. E. Adler, A. I. Bobenko and Y. B. Suris, Discrete nonlinear hiperbolic equations. Classification of integrable cases,, Funct. Anal. Appl, 43 (2009), 3. doi: 10.1007/s10688-009-0002-5. [3] V. E. Adler, A. I. Bobenko and Yu. B. Suris, Geometry of Yang-Baxter maps: Pencils of conics and quadrirational mappings,, Comm. Anal. Geom., 12 (2004), 967. doi: 10.4310/CAG.2004.v12.n5.a1. [4] A. I. Bobenko and Yu. B. Suris, Integrable noncommutative equations on quad-graphs. The consistency approach,, Lett. Math. Phys., 61 (2002), 241. doi: 10.1023/A:1021249131979. [5] A. I. Bobenko and Yu. B. Suris, Integrable systems on quad-graphs,, Int. Math. Res. Not., (2002), 573. doi: 10.1155/S1073792802110075. [6] V. Buchstaber, n-valued groups: Theory and applications,, Moscow Mathematical Journal, 6 (2006), 57. [7] V. M. Buchstaber, Functional equations, associated with addition theorems for elliptic functions, and two-valued algebraic groups,, Russian Math. Surv., 45 (1990), 213. doi: 10.1070/RM1990v045n03ABEH002361. [8] V. M. Buchstaber and V. Dragović, Two-valued groups, Kummer varieties and integrable billiards,, preprint, (). [9] V. M. Buchstaber and S. P. Novikov, Formal groups, power systems and Adams operators,, Mat. Sb. (N. S), 84(126) (1971), 81. [10] V. M. Buchstaber and A. P. Veselov, Integrable correspondences and algebraic representations of multivalued groups,, Internat. Math. Res. Notices, 8 (1996), 381. doi: 10.1155/S1073792896000256. [11] G. Darboux, Principes de Géométrie Analytique,, Gauthier-Villars, (1917). [12] V. Dragović, Poncelet-Darboux curves, their complete decomposition and Marden theorem,, Int. Math. Res. Notes, 2011 (2011), 3502. doi: 10.1093/imrn/rnq229. [13] V. Dragović, Generalization and geometrization of the Kowalevski top,, Communications in Math. Phys., 298 (2010), 37. doi: 10.1007/s00220-010-1066-z. [14] V. Dragović and K. Kukić, New examples of systems of the Kowalevski type,, Regular and Chaotic Dynamics, 16 (2011), 484. doi: 10.1134/S1560354711050054. [15] V. Dragović and K. Kukić, Systems of the Kowalevski type and discriminantly separable polynomials,, Regular and Chaotic Dynamics, 19 (2014), 162. doi: 10.1134/S1560354714020026. [16] V. Dragović and M. Radnović, Poncelet Porisms and Beyond,, Springer, (2011). doi: 10.1007/978-3-0348-0015-0. [17] V. Dragović and M. Radnović, Billiard algebra, integrable line congruences and DR-nets,, J. Nonlinear Mathematical Physics, 19 (2012). doi: 10.1142/S1402925112500192. [18] V. V. Golubev, Lectures on the Integration of Motion of a Heavy Rigid Body Around a Fixed Point,, Gostechizdat, (1953). [19] S. Kowalevski, Sur la probleme de la rotation d'un corps solide autour d'un point fixe,, Acta Math., 12 (1889), 177. doi: 10.1007/BF02592182. [20] J. G. Semple and G. T. Kneebone, Algebraic Projective Geometry,, Clarendon Press, (1998).
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