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.
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