A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form

Table 1

Nine cases of the intersection between the conic and the conic .

Case	Nature of the common point
1	Four real simple points
2	Two pairs of complex conjugate simple points
3	Two real simple points + a pair of complex conjugate simple points
4	Two real simple points + a real double point (simple-contact)
5	A pair of complex conjugate simple points + a real double point (simple-contact)
6	Two real double points (double contact)
7	A pair of complex conjugate double points (double contact)
8	A real simple point + a triple point (three-point contact)
9	A quadruple point (four-point contact)