f := x^4-c1*x^3+c2*x^2-c3*x+c4;

restart;

ferrai := -x1^3*x2*x3*x4-x1^2*x2^2*x3^2-x1^2*x2^2*x4^2-x1^2*x3^2*x4^2-x1*x2^3*x3*x4-x1*x2*x3^3*x4-x1*x2*x3*x4^3-x2^2*x3^2*x4^2+x1^2*x2*x3*y+x1^2*x2*x4*y+x1^2*x3*x4*y+x1*x2^2*x3*y+x1*x2^2*x4*y+x1*x2*x3^2*y+x1*x2*x4^2*y+x1*x3^2*x4*y+x1*x3*x4^2*y+x2^2*x3*x4*y+x2*x3^2*x4*y+x2*x3*x4^2*y-x1*x2*y^2-x1*x3*y^2-x1*x4*y^2-x2*x3*y^2-x2*x4*y^2-x3*x4*y^2+y^3;

coeff(ferrai, y^3);

coeff(ferrai, y^2);

coeff(ferrai, y);

res := ferrai - coeff(ferrai, y^3)*y^3 - coeff(ferrai, y^2)*y^2 - coeff(ferrai, y)*y;

c2 := -coeff(ferrai, y^2)/coeff(ferrai, y^3);

sys1 := c1*c3 - 4*c4 = coeff(ferrai, y)/coeff(ferrai, y^3);

sys2 := -c3^2-(c1^2)*c4+4*c2*c4 = res;

solve([sys1, sys2],[c1,c3,c4]);

number of equations is not enough, is it possible to find back c1,c3,c4?

though c2 is easy to know