Remove["`*"];
zeroPolynomial[A_] := Module[{n = Length[A], base, m, s},
base = NestList[Dot[A, #] &, IdentityMatrix[n], n^2];
m = Transpose[Flatten /@ base];
s = NullSpace[m][[1]];
Total[Table[x^i*s[[i]], {i, Length[s]}]]
]
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