As I pointed out, there will surely be no analytical solution. You should expect that with near certainty, as this is the equivalent of a high order polynomial.
I see no error however. Only the indication (from solve, in the form of the results it yields) that it was unable to compute analytical roots.
solve(eqn)
ans =
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 1) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 2) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 3) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 4) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 5) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 6) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 7) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 8) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 9) - 1
1/root(z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z - 1564000/45621, z, 10) - 1
Again, that was completely expected. However, nothing stops you from getting approximate roots.
vpa(solve(eqn))
ans =
-1.6661435339699159760690274098319
-0.17803633410337122738678393539001
- 0.82944078945054912679553055169416 + 0.67487342116657470095168018942751i
- 1.2392357796067300876301190456248 - 0.63453525124017787853359333001222i
- 1.5503593102427657212185969586246 - 0.38032263725340143579333271938087i
- 1.5503593102427657212185969586246 + 0.38032263725340143579333271938087i
- 1.2392357796067300876301190456248 + 0.63453525124017787853359333001222i
- 0.82944078945054912679553055169416 - 0.67487342116657470095168018942751i
- 0.44428946799323473628513677400307 - 0.47475279125570655866734145566345i
- 0.44428946799323473628513677400307 + 0.47475279125570655866734145566345i
