Comparison of critical behaviors of elliptic and hyperbolic quadratic algebraic equations with variable coefficients

Abstract

A comparison of the behaviours of the elliptic with those of hyperbolic quadratic algebraic equations (QAEs) with free and linear variable coefficients, in vicinity of their critical surfaces is made. The critic values of the elliptic and hyperbolic QAEs with variables coefficients are obtained by can-celling their great determinant. If only the free term of a QAE is variable from -∞ to + ∞ and the QAE are two-dimensional, an elliptic QAE is represented by coaxial ellipses, which decrease in size and collapse in their common centre. A hyperbolic QAE is represented by coaxial hyperbolas, which approach their asymptotes, degenerate in them, jump over them and go away from them. The real solutions of hyperbolic QAEs exist for all the values of free term and for elliptic QAE, if the value of the free term is greater than the critical one, the real solutions of elliptic QAEs do no longer exist. If, additionally, also the free term is variable, critical parabolas occur, if a plane of coefficients is used. The real solutions for elliptic QAE collapse along their critical parabola and do not exist inside of it. The hyperbolic QAE is represented by coaxial hyperbolas which degenerate in their asymptotes and jump over them along their critical parabola.

Keywords

• Qualitative analysis of quadratic algebraic equations (QAEs) with variable coefficients
• Collapses and holes of the real solutions of elliptic QAEs
• Degeneration