Do you roll for Gen Synthesis to see if your ground forces return on a planet in a home system during or after the battle? For example: My home system has one planet in it and it is being invaded, every time my enemy destroys a GF do I roll after I take my casualties and put them back on in my home system? Effectively coming back from the dead and being able to fight in the next combat round? Or do they just die and if I win the combat I roll to see if they come back ?

From the FAQ, page 6:

Q: If a player with the Gen Synthesis technology is attacked
on his only Home System planet, do his troops regenerate on
that planet during combat? Can they fight in that combat?
A: Yes. If the player rolls 5+ for a destroyed Ground Force unit,
that unit would remain on the Home System planet and continue
to participate in the combat.