What are the implications of "u'Fen Tisa Einecha ha'Hashamaimah"?
What is the meaning of "Asher Chalak Hashem l'Chol ha'Amim"?
Rashi #1 and Rashbam #1: It means that Hashem gave the luminaries to the nations to illuminate the world on their behalf.
Rashi #2 and Rashbam #2: It means that He did not prevent them from straying after (worshipping) them as gods. 1
Ramban: With reference to the stars and the constellations, it refers to the 'angel' of each and every nation, which the nations subsequently adopted as gods and worshipped. 2
Rashi: On the contrary, He allowed them to stumble with their nonsense on order to drive them out of the world, as the Pasuk intimates in Tehilim, 26:3. Rashbam - but not Yisrael, whom He took out of Egypt in order to be their G-d, as the Pasuk goes on to explain
Ramban: Which is why the Torah writes in Pasuk 20 "But you Hashem took (for Himself)", implying that Yisrael are the nation of Hashem, who do not have (who do not need) a celestial power to lead them. See Ramban who elaborates.