Some multigrades

Notation: a_1, ..., a_n =^k b_1, ..., b_n means that them sum from i=1 to n of a_i^j and the sum from i=1 to n of b_i^j agree for all j between 1 and k.

1st example:
-26e^2+8e+5, 19e^2-20e-4, -21e^2-12e+4, 4e^2+12e-5, 24e^2+12e-1 =^4 -26e^2-8e+5, 19e^2+20e-4, -21e^2+12e+4, 4e^2-12e-5, 24e^2-12e-1

Numerical: plug in e = 1/2 above, after an affine transformation get
0, 18, 36, 59, 83 =^4 3, 11, 48, 50, 84.

2nd example:
f_1+g_1, f_2+g_2, f_3+g_3, f_4+g_4, f_5+g_5 =^4 f_1-g_1, f_2-g_2, f_3-g_3, f_4-g_4, f_5-g_5

where

f_1 = (78*e^6 - 110*e^4 - 70*e^2 + 6)*s^2 + (135*e^6 - 171*e^4 - 207*e^2 + 19)*s*t + (54*e^4 - 132*e^2 + 14)*t^2
f_2 = (-57*e^6 + 115*e^4 + 5*e^2 + 1)*s^2 + (-135*e^6 + 279*e^4 - 57*e^2 + 9)*s*t + (54*e^4 - 132*e^2 + 14)*t^2
f_3 = (63*e^6 - 45*e^4 + 45*e^2 + 1)*s^2 + (135*e^6 - 351*e^4 + 153*e^2 - 1)*s*t + (-486*e^4 + 108*e^2 - 6)*t^2
f_4 = (-72*e^6 - 180*e^4 - 4)*s^2 + (-135*e^6 - 621*e^4 + 63*e^2 - 11)*s*t + (-486*e^4 + 108*e^2 - 6)*t^2
f_5 = (-12*e^6 + 220*e^4 + 20*e^2 - 4)*s^2 + (864*e^4 + 48*e^2 - 16)*s*t + (864*e^4 + 48*e^2 - 16)*t^2

g_1 = (-120*e^5 - 40*e^3)*s^2 + (-630*e^5 - 180*e^3 + 10*e)*s*t + (-810*e^5 - 180*e^3 + 30*e)*t^2
g_2 = (300*e^5 + 40*e^3 - 20*e)*s^2 + (990*e^5 + 180*e^3 - 50*e)*s*t + (810*e^5 + 180*e^3 - 30*e)*t^2
g_3 = (180*e^5 - 200*e^3 + 20*e)*s^2 + (810*e^5 - 900*e^3 + 90*e)*s*t + (810*e^5 - 900*e^3 + 90*e)*t^2
g_4 = (-180*e^5 + 200*e^3 - 20*e)*s^2 + (-810*e^5 + 900*e^3 - 90*e)*s*t + (-810*e^5 + 900*e^3 - 90*e)*t^2
g_5 = (-180*e^5 + 20*e)*s^2 + (-360*e^5 + 40*e)*s*t

I found this example as part of a Caltech SURF project. Writeups of the main idea behind what I did are here and here, but the hard parts of the project (namely, going from first order deformations to higher order deformations, and going from formal power series solving the problem to polynomials solving the problem using LLL tricks) were never completely written up. Some pretty pictures of families of polynomials satisfying my combinatorial criteria for deformability can be found here.