The problem concerns four variables a,b,c,d to be interpreted as centre (c,d) and principal semiaxes a,b of an ellipse [EQUATION] We wish to know when E lies inside the unit disk D: x 2 + y 2 ≤ 1. More precisely, we seek a set of polynomials {P j (a 2 , b 2 , c 2 , d 2 ,)} for j=1,2,...,n with the property that E≤Dif and only if all P j (a 2 , b 2 , c 2 , d 2 ,) ≤0. It is known that n> 1, but n should be minimal.