The discriminant of quasi m-boundary singularities

We describe the discriminant of deformations of simple quasi m-boundary equivalence classes for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> m≥ 2 . All quasi simple <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> </m:math> m -boundary classes are right equivalent to Arnold’s singularities (ADE). Consequently, their respective discriminants are cylinders over the standard ones, together with some new descriptions involving either submanifolds or generalized Whitney umbrella. Our main results are as follows: (1) Theorem 3.5 on the bifurcation diagram and caustic of quasi <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">A</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:math> {{A}}ₖ singularities. (2) Theorem 3.9 on the bifurcation diagram and caustic of quasi <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">D</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msubsup> </m:math> {{D}}₄⁺ singularity. (3) Theorem 3.10 on the bifurcation diagram and caustic of quasi <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">D</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> <m:mrow> <m:mo>−</m:mo> </m:mrow> </m:msubsup> </m:math> {{D}}₄⁻ singularity. (4) Theorem 3.11 on the bifurcation diagram and caustic of quasi <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:math> {{H}}p,k singularities. The subsequent section provides a concise overview of the primary results.