Number Theory Preprints
Todd Cochrane
New Papers (2008-2010):
Exponential sums over
subgroups of $\mathbb Z_p^*$, with Chris Pinner (2010),
pdf, dvi
Bounds on fewnomial
exponential sums over $\mathbb Z_p^*$, with Chris
Pinner (2010), pdf, dvi
Sum-product
estimates applied to
Waring's problem over finite fields, with J. Cipra,
(2010), pdf, dvi
Small values of Waring's number
mod p, with C. Pinner, (2010), pdf, dvi
Small zeros of quadratic forms
mod p^2, with Ali Hakami (2010),
pdf, dvi
On the parity of k-th powers mod
p, a generalization of a problem of Lehmer, with ...
(2009) pdf, dvi
Explicit bounds on monomial and
binomial exponential sums, with
C. Pinner, (2009) pdf, dvi
The congruence
$x_1x_2=x_3x_4 \pmod m$ and mean values of character
sums, with S. Shi, (2009) pdf,
dvi
Decimations
of $l$-sequences and
permutations of even residues mod $p$, with J. Bourgain, J.
Paulhus and C. Pinner,
SIAM
J. Discrete Math. 23 (2009), no. 2, 842-857. dvi, pdf
Waring's number mod
$m$, with A. Alnaser,
J.
Number Theory 128 (2008), no. 9, 2582-2590. pdf, dvi
Sum-product
estimates applied to
Waring's problem mod $p$, with C. Pinner,
Integers
8
(2008), A46, 18pp. pdf, dvi
Monographs:
I) Small Solutions of Congruences, Ph.D. Thesis, University of Michigan,
1984, 1-106. (Advisor: D.J. Lewis) pdf
II) Exponential Sums and the
Distribution of Solutions of Congruences, Inst. of Math., Academia Sinica,
Taipei, Taiwan (1994), 1-84 pdf, dvi
Additive Combinatorics:
1) Sum-product
estimates applied to
Waring's problem mod $p$.
Integers
8
(2008), A46, 18pp. pdf, dvi
2) Sum-product estimates
applied to Waring's problem over finite fields, with J. Cipra,
(2010), pdf, dvi
3) Small values of Waring's
number mod p, with C. Pinner, (2010), pdf, dvi
Distribution of Solutions of
Congruences:
1) Small Solutions of
Congruences, Ph.D. Thesis, University of Michigan,
1984, Advisor: D.J. Lewis. pdf
2) The
distribution of solutions to equations over finite fields, Trans.
Amer. Math. Soc. 293 (1986), no. 2 819-826. pdf, dvi
3) Small solutions of
congruences over algebraic number fields,
Illinois J. Math. 31 (1987), no. 4, 618-625. pdf, dvi
4) Small
solutions of cubic congruences, Proc. Amer. Math. Soc.
106 (1989), no. 2, 333-334. pdf, dvi
5) Exponential Sums and the
Distribution of Solutions of Congruences, Institute of
Mathematics, Academia Sinica, Taipei, Taiwan (1994), 1-84, pdf, dvi
Exponential Sums over finite
fields:
1)
Exponential Sums and the
Distribution of Solutions of Congruences, Inst. of Math., Academia Sinica,
Taipei, Taiwan (1994), 1-84 pdf, dvi
2) Bounds
on complete exponential sums,
Analytic number theory, Vol. 1 (Allerton Park, IL, 1995),
211-224, Progr.
Math., 138,
Birkhauser Boston,
Boston, MA, 1996. pdf, dvi
3) Bounds
on exponential sums and the
polynomial Waring problem mod $p$ with Chris Pinner and
Jason Rosenhouse,
J. London Math. Soc. (2) 67 (2003),
no. 2,
319-336. pdf, dvi
4) An improved
Mordell type bound for
exponential sums, with Chris Pinner,
Proc. Amer. Math. Soc. 133 (2005), no. 2, 313-320.
pdf, dvi
5) A
further refinement of Mordell's
bound on exponential sums, with Jeremy Coffelt and Chris
Pinner,
Acta Arith. 116 (2005), no. 1, 35-41. pdf, dvi
6) Stepanov's
method applied to
binomial exponential sums,
with Chris Pinner,
Q. J. Math. 54 (2003), no. 3, 243-255. pdf, dvi
7) Using
Stepanov's method for
exponential sums involving rational functions, with Chris Pinner,
J. Number Theory 116 (2006), no. 2, 270-292. pdf, dvi
8) A system of
simultaneous
congruences arising from trinomial exponential sums, with Jeremy Coffelt and Chris Pinner
J. Theor. Nombres Bordeaux
18 (2006), no. 1, 59-72,
pdf, dvi
Exponential Sums modulo prime
powers:
1) Pure and mixed
exponential sums, with Zhiyong
Zheng,
Acta Arith. 91 (1999), no. 3, 249-278. pdf,
dvi
2) Exponential sums with
rational
function entries,
Acta Arith. 95 (2000), no. 1, 67-95. pdf,
dvi
3) On upper bounds of Chalk and Hua for
exponential sums, with
Z. Zheng, Proc.
Amer.
Math. Soc. 129, no. 9, (2001), 2505-2516. pdf,
dvi
4) Bounds
for certain exponential sums, with Z. Zheng,
Asian J. Math. 4 (2000), no. 4, 757-773. pdf,
dvi
5) Upper bounds on $n$-dimensional
Kloosterman sums, with M-C Liu and Z. Zheng,
J.
Number Theory 106 (2004), no. 2, 259-274. pdf,
dvi
6) Upper bounds on a two-term exponential
sum, with Z. Zheng,
Sci.
China Ser. A 44 (2001), no. 8, 1003-1015. pdf,
dvi
7) Exponential sums
modulo prime powers,
Acta
Arith. 101 (2002), no. 2, 131-149. pdf,
dvi
8) A survey on
pure and mixed exponential
sums modulo prime powers, with Z. Zheng,
Number theory for the millennium, I (Urbana, IL, 2000),
273-300, A K
Peters, Natick,MA, 2002. dvi, pdf
9) Sparse
polynomial exponential sums, with J. Rosenhouse and C.
Pinner,
Acta
Arith. 108 (2003), no. 1, 37-52. pdf,
dvi
Multiplicative Character Sums:
1) An upper bound
for character sums, (1987)
preprint, pdf
1) The
congruence $x\sb 1x\sb 2\equiv
x\sb 3x\sb 4\pmod p$, the equation $x\sb 1x\sb 2=x\sb 3x\sb 4$, and
mean values of character sums, with
A . Ayyad and Z. Zheng,
J.
Number Theory 59 (1996), no. 2, 398-413. pdf,
dvi
2) High order moments of
character sums, with Z. Zheng,
Proc. Amer. Math. Soc. 126 no. 4 (1998), 951-956. pdf,
dvi
3) On a problem of H. Cohn for character
sums, with D.
Garth and Z. Zhiyong, J. Number
Theory
81 (2000), 120-129. pdf, dvi
4) Upper bounds on character sums with
rational function entries, with C.L. Liu and Z. Zheng,
Acta
Math. Sin. (Engl. Ser.) 19 (2003), no. 2, 327-338.
pdf,
dvi
Quadratic Forms:
1) Small zeros of quadratic
forms modulo p, J. Number Theory
33 no. 3 (1989), 286-292, pdf, dvi
2) Small zeros of quadratic
forms modulo p, II,Proc.
Illinois Number Theory Conference of 1989, Birkhauser, Boston (1990),
91-94. pdf, dvi
3) Small zeros of quadratic
forms modulo p, III, J. Number
Theory 37 no. 1 (1991), 92-99, pdf, dvi
4) Small zeros of quadratic
congruences modulo pq,
Mathematika 37 (1990), 261-272, pdf, dvi
5) Small zeros of quadratic
congruences modulo pq, II, J.
Number Theory 50 no. 2 (1995), 299-308. pdf,
dvi
6) On representing the multiple
of a number by a quadratic form, Acta Arith. 63 no. 3 (1993),
211-222, pdf, dvi
7) Small
solutions of the congruence
$a\sb 1x\sp 2\sb 1+a\sb 2x\sp 2\sb 2+a\sb 3x\sp 2\sb 3+a\sb 4x\sp 2\sb
4\equiv c\pmod p$, with Z. Zheng,
Acta
Math. Sinica (N.S.) 14 (1998), no. 2, 175-182. pdf, dvi
8) Small solutions of the
Legendre equation, with
P.
Mitchell, J. Number Theory 70 no. 1 (1998), 62-66, pdf, dvi
9) Lattices
in $\Bbb Z\sp 2$ and the
congruence $xy+uv\equiv c\pmod m$, with A. Ayyad,
Acta
Arith. 132 (2008), no. 2, 127--133. pdf,
dvi
10) The
congruence $x\sb 1x\sb 2\equiv
x\sb 3x\sb 4\pmod p$, the equation $x\sb 1x\sb 2=x\sb 3x\sb 4$, and
mean values of character sums, with
A . Ayyad and Z. Zheng,
J.
Number Theory 59 (1996), no. 2, 398-413. pdf,
dvi
11) The congruence
$x_1x_2=x_3x_4 \pmod m$ and mean values of character
sums, with S. Shi, (2009) pdf,
dvi
Trigonometric Sums:
1) On a trigonometric
inequality of Vinogradov, J. Number
Theory 26 no. 1 (1987), 9-16. pdf, dvi
2) An asymptotic formula for a
trigonometric inequality of Vinogradov, with J.C. Peral,
J. Number Theory 91 (2001), 1-19. pdf, dvi
3) Trigonometric approximation
and uniform distribution modulo one, Proc. of the A.M.S. 103 no.
3 (1989), 695-702. pdf, dvi
Uniform Distribution:
1) Trigonometric approximation and
uniform distribution modulo one, Proc. of the A.M.S. 103 no. 3
(1989), 695-702. pdf, dvi
2)
Decimations of $l$-sequences and
permutations of even residues mod $p$, with J. Bourgain, J.
Paulhus and C. Pinner,
SIAM
J. Discrete Math. 23 (2009), no. 2, 842-857. dvi, pdf
3) Distribution of
$\lambda$-roots of composite moduli, with L. Xia and Z. Zheng,
(2004) preprint, pdf, dvi
4) Distribution of primitive
$\lambda$-roots of composite moduli II, with Z. Zheng, Chinese
Annals of Math.-Series B, 27, no. 5. (2006), 549-552. pdf,dvi
Waring's Problem (mod p), (mod m) and
over finite fields:
1) Bounds
on exponential sums and the
polynomial Waring problem mod $p$ with Chris Pinner and
Jason Rosenhouse,
J. London Math.
Soc. (2) 67 (2003), no. 2,
319-336. pdf, dvi
2) Heilbronn's
conjecture on Waring's
number (mod $p$), with J. Cipra and C. Pinner,
J.
Number Theory 125 (2007), no. 2, 289-297. pdf,
dvi
3) Waring's number mod
$m$, with A. Alnaser,
J.
Number Theory 128 (2008), no. 9, 2582-2590. pdf, dvi
4) Sum-product
estimates applied to
Waring's problem mod $p$.
Integers
8
(2008), A46, 18pp. pdf, dvi
5) Sum-product estimates
applied to Waring's problem over finite fields, with J. Cipra,
(2010), pdf, dvi
6) Small values of Waring's
number mod p, with C. Pinner, (2010), pdf, dvi
Miscellaneous Number Theory:
1) Consecutive
triples of sums of two squares, with R.E.
Dressler, Archiv der Math. 49 (1987), 301-304, pdf,
dvi
2) The
diophantine equation f(x)=g(y), Proc. of the A.M.S.
109 no. 3 (1990), 573-577. pdf, dvi
3)
Strong Riesz sets and polynomials,
with R.E. Dressler, Colloquium Math. 58 (1989), 23-27. pdf, dvi
4)
Covering congruences in higher
dimensions, with G. Myerson,
Rocky Mountain J. of Math. 26 no. 1 (1996), 77-81. pdf, dvi
5) Gaps between integers with
the same prime factors, with
R. Dressler, Math. Comp. 68 no. 225 (1999), 395-40.
pdf, dvi