On the Complex and Real Hessian Polynomials

Table 3

Real Hessian polynomials.

𝑥^2𝑦+𝑦^2+𝑥+𝑎1𝑦+𝑎2=0,𝑎1,𝑎2∈ℝ	𝑥^2𝑦+𝑦^2+𝑦+𝑎3=0,𝑎3∈(−∞,0)∪(0,1/4)
𝑥^2𝑦+𝑦^2−𝑦+𝑎4=0,𝑎4<−3/4	𝑥^2𝑦+𝑦^2−1=0
𝑥^2𝑦+3𝑦=0	𝑥^2𝑦−3𝑦=0
𝑥^2𝑦=0	𝑥^3+𝑎5𝑥−𝑦^2−1=0,𝑎5∈ℝ,
𝑥^3−𝑦^2+𝑎6𝑥+1=0,𝑎6∈ℝ	𝑥^3−𝑦^2+𝑥=0
𝑥^3−𝑦^2−𝑥=0	𝑥^3−𝑦^2=0