MCN-2 Invariants, Homomorphisms, and Anomalies in Polynomials and the 'Fundamental Theorem of Algebra'

This article contributes to the existing literature by: i) proving that the Descartes Sign Rule (as interpreted by most academicians - such as Oehmke (2000) and Osborne (2010)) is wrong; ii) proving that the Fundamental Theorem Of Algebra (FTA) is wrong; iii) explaining how “Root-Calculation” in Algebra is wrong and introducing an alternative method for verifying real and complex roots of a polynomial; iv) solving a six-degree Polynomial equation and a nine-degree Polynomial equation, by introducing new classes of Invariants (“MCN-2 Invariants”) and Homomorphisms. Burrus (2004); Sitton, Burrus, Fox & Treitel (2003); and Lei, Blane & Cooper (1996), had concluded that such higher-order polynomials were impossible to solve. These issues are applicable in nonlinear analysis, evolutionary computation and pattern-analysis – given the discussions in Yannacopoulos, Brindley, Merkin & Pilling (1996); Campos-Canton, Aguirre-Hernandez, Renteria & Gonzalez (2015); Zheng, Takamatsu & Ikeuchi (2010); and Boyer & Goh (2007).