1) MAKING AN ITEM HEFKER FOR ONE DAY
QUESTION: The Gemara quotes a Tosefta which states that a person may make an item Hefker for only one day.
Does the Tosefta mean that even if another person takes possession of the item during that day he must give it back to the original owner at the end of the day, or does it mean that anyone may take possession of that item during that day and own it permanently, but after that day no one may take possession of it?
ANSWER: The ROSH writes that if no one takes possession of the item within the day (or any other time period the original owner stipulated), it automatically reverts back to the possession of the original owner. REBBI AKIVA EIGER (Teshuvos #145) and the RASHASH point out that the Rosh implies that if someone does take possession of the item during that day, it will remain his permanently; the original owner merely limited the time during which others may take possession of it.
They explain in depth how this works. The concept of Hefker is learned from the laws of the Shemitah year. During the Shemitah year, one who takes possession of the ownerless produce becomes the full owner. All situations of Hefker work the same way -- the one who acquires the item becomes the permanent owner. (Rebbi Akiva Eiger, however, remains with a question: if it is true that the laws of Hefker are derived from the laws of Shemitah, why is one entitled to limit the time period during which others may acquire the item? The duration of the Shemitah year does is not subject to variation.)
2) RETRACTING HEFKER
QUESTION: Ula says that the entire Tosefta is in accordance with the view of the Chachamim. Although the Chachamim maintain that once one declares an item Hefker the item is no longer under his control, they agree that if one specifies that the item is Hefker for a limited amount of time (see previous Insight) he has the right to retract the Hefker. Ula explains that the reason why the original owner may change his mind in this case and retract the Hefker is that it is not common for a person to make an item Hefker for a limited amount of time.
Why, though, is the fact that this is an unusual way to make an item Hefker a reason to permit the original owner to retract the Hefker?
ANSWERS:
(a) The RAN answers that Ula means that since the person did not make the item Hefker with the standard formula (which would have implied that the item should be Hefker forever until someone takes possession of it), we assume that he wants to retain partial control of the item. We assume that he does not want to make it the type of Hefker which completely leaves his control, and thus it remains in his possession so that he retains the right to retract the Hefker until someone acquires it. (That is, the Chachamim agree with Rebbi Yosi in this case because of this logic.)
(b) TOSFOS answers that although mid'Oraisa the item is Hefker and no longer under his control, the fact that he made an unusual form of Hefker shows that he does not have full intention to relinquish ownership. Therefore, the Chachamim established that the Hefker does not fully take effect and he may still retract it.

44b----------------------------------------44b

3) HEFKER EXEMPTS PRODUCE FROM MA'ASER
OPINIONS: The Gemara asks, "[If so,] perhaps he will come to separate Ma'aser from produce which is obligated on behalf of produce which is exempt."
On what statement does the Gemara ask this question?
(a) TOSFOS understands that the Gemara questions Ula's approach that the Tosefta follows the opinion of the Chachamim. The Gemara asks that since the Chachamim maintain that mid'Oraisa one cannot retract his Hefker, consequently the produce is exempt from Ma'aser, mid'Oraisa. Although mid'Rabanan the Chachamim give the person the right to change his mind, the produce remains Hefker mid'Oraisa, and that is why it is exempt mid'Oraisa from Ma'aser. For this reason, the Gemara asks that since the Chachamim require him, mid'Rabanan, to separate Ma'aser from this produce, he might come to separate Ma'aser from this produce on behalf of other produce which has a Chiyuv Ma'aser mid'Oraisa.
(b) The RAN explains that the Gemara questions the statement of Reish Lakish, who says that the entire Tosefta is in accordance with the view of Rebbi Yosi. One may not retract his Hefker after three days because of a Takanah d'Rabanan (so that people not think that the Hefker did not take effect).
The Gemara assumes that since, mid'Oraisa, one has the power to retract the Hefker, the Hefker that remains after he attempted to retract is only a Hefker mid'Rabanan. Based on this, the Gemara asks that the owner might think that it is Hefker mid'Oraisa and therefore exempt mid'Oraisa from Ma'aser.
4) HEFKER MID'RABANAN EXEMPTS PRODUCE FROM MA'ASER
QUESTION: The Chachamim have the authority to decide who has the ownership of property (based on the principle of "Hefker Beis Din Hefker"). Since this authority is granted to them by the Torah, their decision is binding even mid'Oraisa. Consequently, once the Chachamim decreed that one may not retract his Hefker, the item should remain Hefker mid'Oraisa! How, then, can produce which one made Hefker and then retracted have a Chiyuv d'Oraisa of Terumos and Ma'aseros?
The RAN answers that the Chachamim chose not to use their power of "Hefker Beis Din" to remove the requirement to separate Terumos and Ma'aseros. Hence, the produce remains Hefker only mid'Rabanan.
However, even if the produce remains Hefker mid'Rabanan, since everyone has the right to take it the produce should still be exempt from Terumos and Ma'aseros! The mere fact that the Chachamim permit everyone to take the produce should automatically render it exempt, even if it is not Hefker mid'Oraisa! (See Yerushalmi, beginning of Ma'aseros, and Tosfos to Bava Kama 28a, DH Zeh).
The NETZIV adds that the Gemara in Rosh Hashanah (15a) says that an Esrog which began to grow in the Shemitah year and was picked in the following year is exempt from Ma'aser even though there is a doubt whether it has the Kedushah of Shevi'is. TOSFOS (DH Beshleima) explains that although it is not known for certain whether the Esrog has Kedushas Shevi'is, nevertheless it is definitely exempt from Ma'aser since the Chachamim required the owner to make it Hefker. As long as the owner makes the produce Hefker, even if he does so only as a result of a Safek and the Hefker is mid'Rabanan, the produce is still exempt from Ma'aser. Why is the case of the Gemara here different?
ANSWER: The NETZIV answers that the Ran agrees that if someone else takes possession of the produce that was made Hefker (even though it is only Hefker mid'Rabanan), the produce has no Chiyuv d'Oraisa of Terumos and Ma'aseros since he had permission to take it as a result of the Chachamim's Hefker. When the Ran writes that there is a Chiyuv mid'Oraisa of Terumos and Ma'aseros, he is referring only to a case in which the original owner took the produce for himself. Although the Chachamim removed his right to retract his Hefker, they did so only insofar that he may not change his mind about the Hefker. He may, however, make a new physical acquisition of the produce for himself. In such a case, the fact that Chachamim made it Hefker is irrelevant; he anyway re-takes possession of his produce. Therefore, the case may be treated as one in which he retracted his Hefker, and therefore the produce is obligated in Terumos and Ma'aseros. (See the Netziv for further discussion. See also KOVETZ SHI'URIM 2:14 for another explanation of the RAN.)