Converting Inequalities to Equalities

Converting Inequalities to Equalities

All strict and unstrict inequalities can be written as existence theorems on equalities:

x!=0  <==>  ∃y{xy+1 =0}
x>0   <==>  ∃y{xy²-1=0}   
x<0   <==>  ∃y{xy²+1=0}
x>=0  <==>  ∃y{x-y² =0}  
x<=0  <==>  ∃y{x+y² =0}

Note that the first is the same as ∃y{y=1/x} which is true as long as we don't let x be infinite!

Two equalities can be combined to a single equality like this:

x=0 ∧ y=0  <==>  x²+y² = 0

A polynomial in a high degree can be written as a system in a lower degree:

ax^6+bx^5+cx^4+dx^3+ex²+f = 0 <==>  y-x²=0 ∧ z-y²=0 ∧ azy+bzx+cz+dxy+ey+f = 0
<==> (y-x^2)² +(z-y^2)² +(azy+bzx+cz+dxy+ey+f)² = 0


Hence it turns out that any system of polynomials equalities and inequalities and can be written as a quartic in multiple variables.