sage: R.=QQ[] sage: Reps.=R[] sage: Rxyz.=Reps[] sage: conic=(a*x+b*y+c*z)^2+eps*(A*x^2+B*x*y+C*y^2+D*x*z+E*y*z+F*z^ ....: 2) sage: conic (A*eps + a^2)*x^2 + (B*eps + 2*a*b)*x*y + (C*eps + b^2)*y^2 + (D*eps + 2*a*c)*x*z + (E*eps + 2*b*c)*y*z + (F*eps + c^2)*z^2 sage: Rst.=Reps[] sage: circle=(s^2-t^2,2*s*t,s^2+t^2) sage: conic(circle) ((A + D + F)*eps + a^2 + 2*a*c + c^2)*s^4 + ((2*B + 2*E)*eps + 4*a*b + 4*b*c)*s^3*t + ((-2*A + 4*C + 2*F)*eps - 2*a^2 + 4*b^2 + 2*c^2)*s^2*t^2 + ((-2*B + 2*E)*eps - 4*a*b + 4*b*c)*s*t^3 + ((A - D + F)*eps + a^2 - 2*a*c + c^2)*t^4 sage: RX.=Reps[] sage: conic(circle)(X,1) ((A + D + F)*eps + a^2 + 2*a*c + c^2)*X^4 + ((2*B + 2*E)*eps + 4*a*b + 4*b*c)*X^3 + ((-2*A + 4*C + 2*F)*eps - 2*a^2 + 4*b^2 + 2*c^2)*X^2 + ((-2*B + 2*E)*eps - 4*a*b + 4*b*c)*X + (A - D + F)*eps + a^2 - 2*a*c + c^2 sage: disc=conic(circle)(X,1).discriminant() sage: disc.parent() Univariate Polynomial Ring in eps over Multivariate Polynomial Ring in a, b, c, A, B, C, D, E, F over Rational Field sage: disc.degree() 6 sage: disc[0] 0 sage: disc[1] 0 sage: disc[2] 4096*a^4*b^4*A^2 + 8192*a^2*b^6*A^2 + 4096*b^8*A^2 - 8192*a^4*b^2*c^2*A^2 - 24576*a^2*b^4*c^2*A^2 - 16384*b^6*c^2*A^2 + 4096*a^4*c^4*A^2 + 24576*a^2*b^2*c^4*A^2 + 24576*b^4*c^4*A^2 - 8192*a^2*c^6*A^2 - 16384*b^2*c^6*A^2 + 4096*c^8*A^2 - 8192*a^5*b^3*A*B - 16384*a^3*b^5*A*B - 8192*a*b^7*A*B + 8192*a^5*b*c^2*A*B + 32768*a^3*b^3*c^2*A*B + 24576*a*b^5*c^2*A*B - 16384*a^3*b*c^4*A*B - 24576*a*b^3*c^4*A*B + 8192*a*b*c^6*A*B + 4096*a^6*b^2*B^2 + 8192*a^4*b^4*B^2 + 4096*a^2*b^6*B^2 - 4096*a^6*c^2*B^2 - 20480*a^4*b^2*c^2*B^2 - 20480*a^2*b^4*c^2*B^2 - 4096*b^6*c^2*B^2 + 12288*a^4*c^4*B^2 + 28672*a^2*b^2*c^4*B^2 + 12288*b^4*c^4*B^2 - 12288*a^2*c^6*B^2 - 12288*b^2*c^6*B^2 + 4096*c^8*B^2 + 8192*a^6*b^2*A*C + 16384*a^4*b^4*A*C + 8192*a^2*b^6*A*C + 8192*a^6*c^2*A*C + 8192*a^4*b^2*c^2*A*C + 8192*a^2*b^4*c^2*A*C + 8192*b^6*c^2*A*C - 24576*a^4*c^4*A*C - 40960*a^2*b^2*c^4*A*C - 24576*b^4*c^4*A*C + 24576*a^2*c^6*A*C + 24576*b^2*c^6*A*C - 8192*c^8*A*C - 8192*a^7*b*B*C - 16384*a^5*b^3*B*C - 8192*a^3*b^5*B*C + 24576*a^5*b*c^2*B*C + 32768*a^3*b^3*c^2*B*C + 8192*a*b^5*c^2*B*C - 24576*a^3*b*c^4*B*C - 16384*a*b^3*c^4*B*C + 8192*a*b*c^6*B*C + 4096*a^8*C^2 + 8192*a^6*b^2*C^2 + 4096*a^4*b^4*C^2 - 16384*a^6*c^2*C^2 - 24576*a^4*b^2*c^2*C^2 - 8192*a^2*b^4*c^2*C^2 + 24576*a^4*c^4*C^2 + 24576*a^2*b^2*c^4*C^2 + 4096*b^4*c^4*C^2 - 16384*a^2*c^6*C^2 - 8192*b^2*c^6*C^2 + 4096*c^8*C^2 + 8192*a^5*b^2*c*A*D + 16384*a^3*b^4*c*A*D + 8192*a*b^6*c*A*D - 8192*a^5*c^3*A*D - 32768*a^3*b^2*c^3*A*D - 24576*a*b^4*c^3*A*D + 16384*a^3*c^5*A*D + 24576*a*b^2*c^5*A*D - 8192*a*c^7*A*D + 8192*a^4*b^3*c*B*D + 16384*a^2*b^5*c*B*D + 8192*b^7*c*B*D - 8192*a^4*b*c^3*B*D - 32768*a^2*b^3*c^3*B*D - 24576*b^5*c^3*B*D + 16384*a^2*b*c^5*B*D + 24576*b^3*c^5*B*D - 8192*b*c^7*B*D - 8192*a^7*c*C*D - 32768*a^5*b^2*c*C*D - 40960*a^3*b^4*c*C*D - 16384*a*b^6*c*C*D + 24576*a^5*c^3*C*D + 65536*a^3*b^2*c^3*C*D + 40960*a*b^4*c^3*C*D - 24576*a^3*c^5*C*D - 32768*a*b^2*c^5*C*D + 8192*a*c^7*C*D - 4096*a^6*b^2*D^2 - 12288*a^4*b^4*D^2 - 12288*a^2*b^6*D^2 - 4096*b^8*D^2 + 4096*a^6*c^2*D^2 + 20480*a^4*b^2*c^2*D^2 + 28672*a^2*b^4*c^2*D^2 + 12288*b^6*c^2*D^2 - 8192*a^4*c^4*D^2 - 20480*a^2*b^2*c^4*D^2 - 12288*b^4*c^4*D^2 + 4096*a^2*c^6*D^2 + 4096*b^2*c^6*D^2 - 16384*a^6*b*c*A*E - 40960*a^4*b^3*c*A*E - 32768*a^2*b^5*c*A*E - 8192*b^7*c*A*E + 40960*a^4*b*c^3*A*E + 65536*a^2*b^3*c^3*A*E + 24576*b^5*c^3*A*E - 32768*a^2*b*c^5*A*E - 24576*b^3*c^5*A*E + 8192*b*c^7*A*E + 8192*a^7*c*B*E + 16384*a^5*b^2*c*B*E + 8192*a^3*b^4*c*B*E - 24576*a^5*c^3*B*E - 32768*a^3*b^2*c^3*B*E - 8192*a*b^4*c^3*B*E + 24576*a^3*c^5*B*E + 16384*a*b^2*c^5*B*E - 8192*a*c^7*B*E + 8192*a^6*b*c*C*E + 16384*a^4*b^3*c*C*E + 8192*a^2*b^5*c*C*E - 24576*a^4*b*c^3*C*E - 32768*a^2*b^3*c^3*C*E - 8192*b^5*c^3*C*E + 24576*a^2*b*c^5*C*E + 16384*b^3*c^5*C*E - 8192*b*c^7*C*E + 8192*a^7*b*D*E + 24576*a^5*b^3*D*E + 24576*a^3*b^5*D*E + 8192*a*b^7*D*E - 16384*a^5*b*c^2*D*E - 32768*a^3*b^3*c^2*D*E - 16384*a*b^5*c^2*D*E + 8192*a^3*b*c^4*D*E + 8192*a*b^3*c^4*D*E - 4096*a^8*E^2 - 12288*a^6*b^2*E^2 - 12288*a^4*b^4*E^2 - 4096*a^2*b^6*E^2 + 12288*a^6*c^2*E^2 + 28672*a^4*b^2*c^2*E^2 + 20480*a^2*b^4*c^2*E^2 + 4096*b^6*c^2*E^2 - 12288*a^4*c^4*E^2 - 20480*a^2*b^2*c^4*E^2 - 8192*b^4*c^4*E^2 + 4096*a^2*c^6*E^2 + 4096*b^2*c^6*E^2 + 8192*a^6*b^2*A*F + 24576*a^4*b^4*A*F + 24576*a^2*b^6*A*F + 8192*b^8*A*F + 8192*a^6*c^2*A*F - 8192*a^4*b^2*c^2*A*F - 40960*a^2*b^4*c^2*A*F - 24576*b^6*c^2*A*F - 16384*a^4*c^4*A*F + 8192*a^2*b^2*c^4*A*F + 24576*b^4*c^4*A*F + 8192*a^2*c^6*A*F - 8192*b^2*c^6*A*F - 8192*a^7*b*B*F - 24576*a^5*b^3*B*F - 24576*a^3*b^5*B*F - 8192*a*b^7*B*F + 32768*a^5*b*c^2*B*F + 65536*a^3*b^3*c^2*B*F + 32768*a*b^5*c^2*B*F - 40960*a^3*b*c^4*B*F - 40960*a*b^3*c^4*B*F + 16384*a*b*c^6*B*F + 8192*a^8*C*F + 24576*a^6*b^2*C*F + 24576*a^4*b^4*C*F + 8192*a^2*b^6*C*F - 24576*a^6*c^2*C*F - 40960*a^4*b^2*c^2*C*F - 8192*a^2*b^4*c^2*C*F + 8192*b^6*c^2*C*F + 24576*a^4*c^4*C*F + 8192*a^2*b^2*c^4*C*F - 16384*b^4*c^4*C*F - 8192*a^2*c^6*C*F + 8192*b^2*c^6*C*F - 8192*a^7*c*D*F - 24576*a^5*b^2*c*D*F - 24576*a^3*b^4*c*D*F - 8192*a*b^6*c*D*F + 16384*a^5*c^3*D*F + 32768*a^3*b^2*c^3*D*F + 16384*a*b^4*c^ sage: Rh.=PowerSeriesRing(QQ) sage: (1+2*h)^6/(1+h)^3 1 + 9*h + 30*h^2 + 42*h^3 + 15*h^4 - 9*h^5 + 4*h^6 - 3*h^8 + 5*h^9 - 6*h^10 + 6*h^11 - 5*h^12 + 3*h^13 - 4*h^15 + 9*h^16 - 15*h^17 + 22*h^18 - 30*h^19 + O(h^20) sage: 3^5-90*4+15*18-51 102 sage: (3^5-90*4+15*18-51)*32 3264