On the Complex and Real Hessian Polynomials

Table 4

Real polynomials which are not Hessian polynomials.

𝑥𝑦^2−𝑥(𝑥−3)^2+𝑏1𝑥+𝑏2𝑦−𝑏3=0,𝑏1,𝑏2,𝑏3∈ℝ	𝑥𝑦^2−𝑥^3+𝑏4𝑥+2𝑦−𝑏5=0,𝑏5∈ℝ
𝑥𝑦^2−𝑥^3+𝑏6𝑥−1=0,𝑏6∈ℝ	𝑥𝑦^2−𝑥^3+𝑥=0
𝑥𝑦^2−𝑥^3−𝑥=0	𝑥𝑦^2−𝑥^3=0
𝑥^3+𝑥𝑦^2−6𝑦^2+𝑏7𝑥+𝑏8𝑦+𝑏9=0,𝑏7,𝑏8,𝑏9∈ℝ	𝑥^3+𝑥𝑦^2+𝑏10𝑥+3𝑦+𝑏11=0,𝑏10,𝑏11∈ℝ
𝑥^3+𝑥𝑦^2+𝑏12𝑥+1=0,𝑏12∈ℝ	𝑥^3+𝑥𝑦^2+3𝑥=0
𝑥^3+𝑥𝑦^2−3𝑥=0	𝑥^3+𝑥𝑦^2=0
𝑥^2𝑦+𝑦^2+𝑦+𝑐=0,𝑐∈{0}∪[1/4,∞)	𝑥^2𝑦+𝑦^2−𝑦+𝑑=0,𝑑≥−3/4
𝑥^2𝑦+𝑦^2+1=0	𝑥^2𝑦+𝑦^2=0
𝑥^2𝑦+3𝑥+3𝑦+𝑏13=0,𝑏13∈ℝ	𝑥^2𝑦+3𝑦+1=0
𝑥^2𝑦+3𝑥−3𝑦+𝑏14=0,𝑏14∈ℝ	𝑥^2𝑦−3𝑦+1=0
𝑥^2𝑦+3𝑥+1=0	𝑥^2𝑦+3𝑥=0
𝑥^2𝑦+1=0	𝑥^3−𝑥𝑦+1=0
𝑥^3−𝑥𝑦=0	𝑥^3−𝑦=0
𝑥^3+𝑏15𝑥+1=0,𝑏15∈ℝ	𝑥^3−3𝑥=0
𝑥^3+3𝑥=0	𝑥^3=0