Gemara

57a (Mishnah): If one said 'these Peros are forbidden to me', he may not benefit from Chilufeihem (what they are exchanged for) or Giduleihem (what grows from them).

If he said 'that I will eat' or 'that I will taste', he is permitted Chilufeihem and Giduleihem of a species in which the seed disintegrates. If the seed remains intact (i.e. Chasayos, species like garlic and onions), even Gidulei Giduleihem (what grows from Gidulim) are forbidden.

47a Question (Rami bar Chama): If one said 'this produce is forbidden to Ploni', may Ploni benefit from Chilufeihem (what it is exchanged for)?

Perhaps one cannot forbid to Ploni Davar she'Lo Ba l'Olam, just like he cannot forbid Ploni's property to Ploni;

Or, perhaps Chilufeihem are forbidden, just like Giduleihem!

58b (Beraisa - R. Shimon ben Elazar): If one was weeding in the year after Shemitah with a Yisrael suspected of transgressing Shemitah, he may eat Chasayos

Inference: The added growth in the eighth year nullifies the bulb (of Shemitah)!

59a - Question (Rami bar Chama - Mishnah): If one said 'these Peros are forbidden to me' or 'to my mouth', he may not benefit from Chilufeihem or Giduleihem. If he forbade 'What I eat' or 'what I taste', he is permitted Chilufeihem and Giduleihem;

This applies to a species in which the seed disintegrates. Even Gidulei Giduleihem of Chasayos are forbidden. (The Isur is not Batel in the additional growth!)

Answer (R. Aba): Vows are different. Since he can permit his vow, it is like a Davar she'Yesh Lo Matirim, which is not Batel in a majority of Heter.

Rishonim

Rif (19b): The Gemara concludes that Heter growth is Mevatel Isur, even regarding Chasayos. This does not apply to Nedarim, because Yesh Lo Matirim.

Rambam (Hilchos Nedarim 5:13): If Reuven said 'these Peros are Korban to me' or 'to my mouth', he is forbidden Chilufeihem and Giduleihem.

Rambam (14): If he vowed or swore not to eat or taste them, if the seed disintegrates, such as wheat and barley, Chilufeihem and Giduleihem are permitted. If it is Chasayos, even Gidulei Giduleihem are forbidden.

Rambam (15): Similarly, if a man told his wife 'what you produce is Korban to me' or 'to my mouth', he is forbidden Chilufeihem and Giduleihem. If he said 'I will not eat or taste them', if the seed disintegrates, Chilufeihem and Giduleihem are permitted. If it is Chasayos, even Gidulei Giduleihem are forbidden. The bulb is not Batel in the additional growth because it is Davar she'Yesh Lo Matirim, which is not Batel in a majority.

Rambam (16): If Reuven forbade his produce to Shimon through a Neder or Shevu'ah, it is a Safek whether or not Chilufeihem and Giduleihem are forbidden. Therefore, Shimon is forbidden. If he benefited, he benefited.

Rebuttal (Ra'avad): There was no Safek about Giduleihem, only about Chilufeihem (47a)!

Defense (Or Some'ach 5:11 DH v'Keivan and Oneg Yom Tov YD 86): Rami bar Chama was sure that Giduleihem are forbidden, i.e. of Chasayos. His question from the Mishnah of Konamos (59a) shows that he holds that added growth does not nullify the bulb. We hold that added growth nullifies the bulb. Konamos are different, because Yesh Lo Matirim. However, this is only when he forbids himself. If he forbade another, it is Ein Lo Matirim. Therefore, if a Noder does not intend to forbid Giduleihem, it is permitted to the Mudar.

Ra'avad: I do not know why we ever forbid Giduleihem in a species in which the seed disintegrates. We permit Zeh v'Zeh Gorem (something that results from two causes, one of which is forbidden)!

Answer (Kesef Mishneh): We are stringent about Nedarim, for they are like Hekdesh.

Rosh and Ran (57a DH Masnisin): He is forbidden Giduleihem, just like Gidulei Hekdesh are forbidden.

Question (Mar'eh ha'Panim 25b DH Asur): Gidulei Hekdesh are Chulin, even if he did not redeem them (Terumos 9:4), even regarding Chasayos. The Rambam (Hilchos Me'ilah 5:6) says that Me'ilah applies to fruits that grow from a Hekdesh tree. However, our Gemara discuss other things as well, e.g. species in which the seed disintegrates.

Ran (57a DH she'Ani): Even though 'that I will eat or taste' normally forbids, one who eats Chilufeihem and Giduleihem does not taste what he forbade.

Poskim

Shulchan Aruch (YD 216:1): If one said 'these Peros are Konam to me', he is forbidden Giduleihem and Chilufeihem. Gidulei Giduleihem are forbidden.

Beis Yosef (DH v'Im): The Rambam forbids because it is Yesh Lo Matirim, which is never Batel.

Question (Bach): The Ran forbids Gidulei Giduleihem even when he said 'these Peros are Konam to me.' Why does the Shulchan Aruch rule against the Rambam, Rosh and Tur, who permit?

Answer #1 (Shach 2): The Rambam, Rosh and Tur do not explicitly permit. In the Seifa of their words they forbid Gidulei Giduleihem. We can say that this applies to the Reisha, also!

Answer #2 (Taz 3): The Rambam, Rosh and Tur did not need to explicitly forbid, for we learn from a Kal va'Chomer. If he forbade 'what I eat or taste', in a species in which the seed disintegrates, Chilufeihem and Giduleihem are permitted, but Gidulei Giduleihem of Chasayos are forbidden. If one said 'these Peros are Konam to me', he is forbidden Giduleihem and Chilufeihem in species in which the seed disintegrates, all the more so Gidulei Giduleihem of Chasayos are forbidden!

Shulchan Aruch (ibid.): If the seed disintegrates, Gidulei Giduleihem are permitted. The same applies if he did not say 'these', e.g. they were in front of him and he said 'they are Konam to me.'

Beis Yosef (DH u'Mah she'Chosav Rabeinu): This is like the Rosh. Perhaps the Ran would agree that this is as if he specified (said 'these').

Rema: Similarly, if he said 'Ploni's Peros' or 'Peros of the place Ploni are Konam to me', it is as if he said 'these Peros'.

Shach (4): The Levush says that the same applies if he forbade a species. This is wrong. The Ran (57a DH Masnisin) permits Chilufeihem and Giduleihem in this case.

Taz (3): Really, the Ran permits only Chilufeihem, for he forbade the entire species! However, perhaps the Ran discusses one who forbade what will grow from the species this year.

Shulchan Aruch (2): If he said 'what I eat or taste from them is Konam', if the seed disintegrates, Chilufeihem and Giduleihem are permitted.

Shulchan Aruch (ibid.): If it is Chasayos, even Gidulei Giduleihem are forbidden. Similarly, if he said 'these Peros are forbidden to Ploni', Chilufeihem and Giduleihem are forbidden.

Shach (7): This is just like when he forbids to himself.

Question (Gilyon Maharsha): When he forbids to himself, it is Yesh Lo Matirim. When he forbids to Ploni, it is Ein Lo Matirim!

Answer (Shalmei Nedarim 47a DH b'Ran DH ha'Mekadesh): Since the Rambam forbids Reuven to give the food to Shimon, it is a Mitzvah for Reuven to permit his vow, therefore it is considered Yesh Lo Matirim.