Waring's number for large subgroups of ℤ*ₚ*

Todd Cochrane; Derrick Hart; Christopher Pinner; Craig Spencer

Todd Cochrane ; Derrick Hart ; Christopher Pinner ; Craig Spencer

Acta Arithmetica, Tome 166 (2014), p. 309-325 / Harvested from The Polish Digital Mathematics Library

Résumé

Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t ₄ = p^{22 / 39 + ϵ}$ , $t ₅ = p^{15 / 29 + ϵ}$ and for s ≥ 6, $t ₛ = p^{(9 s + 45) / (29 s + 33) + ϵ}$ . For s ≥ 24 further improvements are made, such as $t_{32} = p^{5 / 16 + ϵ}$ and $t_{128} = p^{1 / 4}$ .

Publié le : 2014-01-01

Zbl 1314.11051

EUDML-ID : urn:eudml:doc:286595

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     author = {Todd Cochrane and Derrick Hart and Christopher Pinner and Craig Spencer},
     title = {Waring's number for large subgroups of Z*p*},
     journal = {Acta Arithmetica},
     volume = {166},
     year = {2014},
     pages = {309-325},
     zbl = {1314.11051},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-4-2}
}

Todd Cochrane; Derrick Hart; Christopher Pinner; Craig Spencer. Waring's number for large subgroups of ℤ*ₚ*. Acta Arithmetica, Tome 166 (2014) pp. 309-325. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-4-2/