From: voloch@math.utexas.edu (Felipe Voloch)
Subject: Re: Unusual citation in Acta Arithmetica
Date: 2 Dec 1999 16:32:10 GMT
Newsgroups: sci.math
Keywords: the p-adic Waring's problem

Nico Benschop (benschop@iae.nl) wrote:
: Gerry Myerson wrote:
: > 
: > I quote from J F Voloch, On the p-adic Waring's problem, 
: > Acta Arith 90 (1999) 91-95:
: > 
: >  My interest in the subject was started by reading a post by 
: >  N. Benschop on the Usenet newsgroup sci.math where he claimed, in 
: >  effect, that g_{Z_p} \le 4 for all p. After overcoming my initial 

Correction: g_{Z_p}(p) \le 4.


: >  disbelief of the statement, through numerical experimentation, I 
: >  looked at Benschop's paper, but the proof there is unfortunately 
: >  incorrect...."
: > 
: > Gerry Myerson (gerry@mpce.mq.edu.au)
: 

Before I continue let me quote the rest of the paragraph that Myerson
is quoting:

"...although he does rediscover part of Bovey's argument. A search through 
MathSciNet then unearthed Bovey's paper, which sparked the present work."

: Interesting. I'm very honored by the citation;-) 
: So he patched-up the proof? 
: Or maybe the result is faulty, and he found a counter example?
: 

The result that every p-adic integer is a sum of at most 4 p-th
powers is correct. The proof is a corollary of Bovey's results
in the paper:

[B] J. D. Bovey, {\it A note on Waring's problem in $p$-adic fields} Acta Arith.
{\bf XXIX} (1976) 343-351.

But actually this particular result was proved even earlier by
Bhaskaran, "Sums of p-th powers in a -adic ring" Acta Arith. XV (1969)217-219.

For the record, my paper has nothing to add on sums of p-th powers
except to remark that for all p up to  211, except p=3,7,11,17,59,
three p-th powers (instead of four) are enough. My paper is concerned
with improving Bovey's bounds for the number of dp-th powers one needs
for d > 1. 

: Unfortunately, I now nothing about his work (I *do* remember
: seeing several visits from U-Texas/Austin on my homepg... with 
:      the prospect of improved versions of my other papers?) 
: I'll have to look it up in another library, since our's does 
: not have AA. I'm curious (yellow;-)

You should definitely consult Acta Arithmetica, not because of my
paper but because of those mentioned above. My paper, by the way,
is available from my web page:

http://www.ma.utexas.edu/~voloch/

: -- 
: Ciao, Nico Benschop  -- http://www.iae.nl/users/benschop


Felipe