PASS V_0 = <5Y^4Z, 5cXY^4 + 10Y^3Z^2>: [5*Y^4*Z, 5*c*X*Y^4 + 10*Y^3*Z^2]  [0.0s]
PASS r = 3 over QQ(t,c); Syz has a polynomial basis independent at t = 0  [0.1s]
PASS (I_Gamma(t))_2 basis vanishes at the three points  [0.1s]
  dim ker mu over QQ(t,c): 1
  k(0,c) nonzero entries: [1, -1/27179089920000000000, -1]
PASS mu_t has a polynomial kernel vector k(t,c) with k(0,c) having a nonzero monomial entry (nonzero for all c != 0)  [0.3s]
   (K) F_1 = 5*Y^4*Z + 10*t^2*Y^2*Z^3 + t^4*Z^5
   (K) F_2 = 81*t^8*c^5*X^5 + 135*t^6*c^4*X^4*Y + 90*t^4*c^3*X^3*Y^2 + 30*t^2*c^2*X^2*Y^3 + 5*c*X*Y^4 + 270*t^7*c^4*X^4*Z + 360*t^5*c^3*X^3*Y*Z + 180*t^3*c^2*X^2*Y^2*Z + 40*t*c*X*Y^3*Z + 360*t^6*c^3*X^3*Z^2 + 360*t^4*c^2*X^2*Y*Z^2 + 120*t^2*c*X*Y^2*Z^2 + 10*Y^3*Z^2 + 240*t^5*c^2*X^2*Z^3 + 160*t^3*c*X*Y*Z^3 + 20*t*Y^2*Z^3 + 80*t^4*c*X*Z^4 + 25*t^2*Y*Z^4 + 10*t^3*Z^5
PASS (K) F_1 = (P(t)^5 - P(-t)^5)/(2t) and F_2 = (2P(2t)^5 - 3P(t)^5 + P(-t)^5)/(6t^2) are polynomial in (t, c), span V_t, with limits 5Y^4Z and 5cXY^4 + 10Y^3Z^2  [0.3s]
   (K) s_1, s_2, s_3 = [(x, 0), ((-2*t*c)*y + (-c)*z, x + (-3*t^2*c)*y), (-y, (-2*t)*y + z)]
PASS (K) s_1 = (x, 0), s_2 = (-c(2ty + z), x - 3ct^2 y), s_3 = (-y, z - 2ty) are syzygies of (F_1, F_2) identically in (t, c) and a basis of Syz(V_t) (dim Syz = 6 - r = 3), independent at t = 0 for every c (a constant 3x3 minor at t = 0: [1, 1])  [0.3s]
PASS (K) g_1 s_3 - g_2 s_2 + g_3 s_1 = (0, 0) identically in QQ[t, c], g_1 = x(x - 3ct^2 y), g_2 = x(z - 2ty), g_3 = xy - cz^2 + ct^2 y^2: a kernel vector of mu_t with constant coefficients (0, 0, 1; 0, -1, 0; 1, 0, 0) in the basis g_i (x) s_j, nonzero at t = 0; times x it lies in K_t  [0.3s]
total time 0.3 s
ALL CHECKS PASSED
