FriCAS does this in no time:
sage: R.<z, y, x, t> = PolynomialRing(QQ)
sage: f = z^40*y^6*x^2100 + 2*t^15*z^53*y^9*x^2078 + z^40*y^7*x^2003 +
z^40*y^6*x^2000 - 14*z^31*y^6*x^1102 - 2*t*z^20*y^3*x^1101 +
78*z^20*y^3*x^1100 - 28*t^15*z^44*y^9*x^1080 - 4*t^16*z^33
....: *y^6*x^1079 + 156*t^15*z^33*y^6*x^1078 - 14*z^31*y^7*x^1005 -
2*t*z^20*y^4*x^1004 + 78*z^20*y^4*x^1003 - 14*z^31*y^6*x^1002 -
2*t*z^20*y^3*x^1001 + 78*z^20*y^3*x^1000 + 49*z^22*y^6*x^1
....: 04 + 14*t*z^11*y^3*x^103 + (-546*z^11*y^3 + t^2)*x^102 - 78*t*x^101 +
1521*x^100 + 98*t^15*z^35*y^9*x^82 + 28*t^16*z^24*y^6*x^81 +
(-1092*t^15*z^24*y^6 + 2*t^17*z^13*y^3)*x^80 - 156*t^
....: 16*z^13*y^3*x^79 + 3042*t^15*z^13*y^3*x^78 + 49*z^22*y^7*x^7 +
14*t*z^11*y^4*x^6 + (-546*z^11*y^4 + t^2*y)*x^5 + (49*z^22*y^6 -
78*t*y)*x^4 + (14*t*z^11*y^3 + 1521*y)*x^3 + (-546*z^11*
....: y^3 + t^2)*x^2 - 78*t*x + 1521
sage: g = 4*x^1156*y^9*z^46*t^30 + 4*x^1178*y^6*z^33*t^15 + x^1200*y^3*z^20
+ 4*x^1081*y^7*z^33*t^15 + 4*x^1078*y^6*z^33*t^15 + 2*x^1103*y^4*z^20 +
2*x^1100*y^3*z^20 + x^1006*y^5*z^20 + 2*x^
....: 1003*y^4*z^20 + x^1000*y^3*z^20 - 28*x^158*y^9*z^37*t^30 -
28*x^180*y^6*z^24*t^15 - 4*x^157*y^6*z^26*t^31 + 156*x^156*y^6*z^26*t^30 -
7*x^202*y^3*z^11 - 4*x^179*y^3*z^13*t^16 + 156*x^1
....: 78*y^3*z^13*t^15 - x^201*t + 39*x^200 - 28*x^83*y^7*z^24*t^15 -
28*x^80*y^6*z^24*t^15 - 14*x^105*y^4*z^11 - 14*x^102*y^3*z^11 -
4*x^82*y^4*z^13*t^16 + 156*x^81*y^4*z^13*t^15 - 4*x^79*y
....: ^3*z^13*t^16 + 156*x^78*y^3*z^13*t^15 - 2*x^104*y*t + 78*x^103*y -
2*x^101*t + 78*x^100 - 7*x^8*y^5*z^11 - 14*x^5*y^4*z^11 - 7*x^2*y^3*z^11 -
x^7*y^2*t + 39*x^6*y^2 - 2*x^4*y*t + 78*x^
....: 3*y - x*t + 39
sage: fricas(g).gcd(fricas(f)).sage()
2*z^33*y^6*x^1078*t^15 + z^20*y^3*x^1100 + z^20*y^4*x^1003 +
z^20*y^3*x^1000 - 14*z^24*y^6*x^80*t^15 - 7*z^11*y^3*x^102 -
2*z^13*y^3*x^79*t^16 + 78*z^13*y^3*x^78*t^15 - x^101*t + 39*x^100 -
7*z^11*y^4*x^5 - 7*z^11*y^3*x^2 - y*x^4*t + 39*y*x^3 - x*t + 39
sage: timeit("fricas(g).gcd(fricas(f)).sage()")
5 loops, best of 3: 329 ms per loop
(and this includes a very stupid way of communication)
Martin
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