Bruce Reznick
Bruce Reznick
Professor
"The John and Harriet J. Absentminded Professor of Mathematics"
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street
Urbana, Illinois 61801-2975
Office: 327
Altgeld Hall
Office Tel: (217)333-4284
Dept. FAX: (217) 333-9576
e-mail:reznick@math.uiuc.edu

Spring 2018 teaching
Math 496

Links relevant to Math 499 talk, 11/27/17
"Resources for research"
"One introduction to mathematical research",
or "You've proved a theorem, now what?"
"Some thoughts on writing for the Putnam"
Preprint
Cantor set arithmetic (with Jayadev S. Athreya and Jeremy T. Tyson)
Revised version
Congruence properties of binary partition functions (with Katherine Anders, Melissa Dennison and Jennifer Weber Lansing) .

Fall 2017 teaching
Math 417
Math 428
CV-type links
Biographical Sketch
List of Research Publications
MathSciNet
listings
List of Other Publications
List of PhD Theses Supervised and Further
Publications from my Students

SIAM-AG17, Atlanta
Moment Problems and Inequalities for Power Sums, August 4, 2017

2017 East Coast Computer Algebra Day
Rainy Day Lemmas #12 and 35 ,
April 29, 2017
Oberwolfach MFO Workshop 1710 -- Real Algebra
Geometry With a View Toward Moment Problems and Optimization
Extended Abstract -- Moment problems and psd symmetric polynomials,
March 10, 2017
2017 Joint Mathematical Meeting, Atlanta
Sums of powers of binary quadratic forms (corrected version),
January 4, 2017
Algebraic Geometry Seminar, Purdue
Video of seminar, October 13, 2016
Beijing Forestry University, May 26, 2016
The secret lives of polynomial identities
Chern Institute of Mathematics, Nankai University -- TMA 2016
Sums of powers of forms (corrected version), May 24, 2016
Boise State University, April 21, 2016
The secret lives of polynomial identities
2016 Spring Sectional Meeting, North Dakota State University
Sums of powers of binary quadratic forms, April 16, 2016
Ramanujan Competition Awards Ceremony, University of Mauritius
Ramanujan Memorial Lecture, March 2, 2016
Luminy --Structures algébrique ordonnées et leurs interactions
Ternary forms with lots of zeros (slightly corrected version),
October 16, 2015
2015 Fall Sectional Meeting, Loyola-Chicago
Quotients of sums of distinct powers of three, October 3, 2015

SIAM-AG15, Daejeon, South Korea
Discrete zeros of real ternary psd forms, August 5, 2015
Waring rank and the underlying field, August 6, 2015

2014 SIAM Conference on Optimization, San Diego
Real psd ternary forms with many zeros, May 21, 2014

2014 Joint Mathematical Meetings, Baltimore
Beamer pages Material related
to the talk "Every math major should know this crazy
theorem."

NUS-IMS Program on Inverse Moment Problem, Beamer talks, Singapore
From Banach spaces to moments to positive polynomials, December 2,
2103
Public lecture: A walk down the algebraic-geometric mean streets of
mathematics, December 17, 2103
Ternary forms with lots of zeros (not a complete beamer talk, mostly
equations and pictures), December 19, 2013.

Eastern Illinois University, October 4, 2013
The secret lives of polynomial identities (slightly corrected)

SIAM-AG13, August 2, 2013, Ft. Collins
On elliptesque and hyperbolesque curves (slightly corrected, twice)
Forms as sums of powers of lower degree forms (slightly corrected,
with a bonus new last page!)

REGS minicourse home page
Summer 2013

Math 499 Beamer Talk, March 29, 2013
The secret lives of polynomial identities

SIAM 2012, July 13, 2012)
Klein's idea and identities for powers of polynomials
Both beamer slides and a recording of the talk with moving slides!

Seattle Beamer Talk, May 18, 2012
The secret lives of polynomial identities My 5/18/12 talk,
with a few corrections.

Chicago Beamer Talk, December 8, 2011
An updated version of the "steampunk canonical forms" talk.

Banff Beamer Talk, November 17, 2011
A shorter version of the seminar talks, with some new stuff

Seminars on November 1, 2011
Longer abstract of my two seminar
talks in number theory and geometry/AGC on November 1, 2011.
Linearly dependent powers of quadratic forms A combination of the
beamer slides for the two seminars.

Algecom5 Beamer Talk, October 29, 2011
Steampunk canonical forms 10/29/11 talk, an evolution of
the
talks of 3/19/11 and 10/7/11.
Raleigh and Steampunk Canonical Form Material
Material relevant to my
presentation at the SIAM Conference on Applied Algebraic Geometry,
10/7/11.

Claremont Beamer Talk
The secret lives of polynomial identities My 9/21/11
talk.

Konstanz Beamer Talk
The secret lives of polynomial identities My 6/30/11 talk,
with a few corrections.

Iowa Beamer Talk
Material relevant to my
presentation at the Iowa City Sectional Meeting, 3/19/11

Math 499 (2010-2011) Talk
The secret lives of polynomial identities

Presentation at the Memorial to Jerry Uhl, held on 11/6/10:
Adaptation of retirement talk given on 5/4/08

Presentation at the Gathering for Gardner, held on 10/21/10:
Thanks from an appreciative reader

Presentation at the Departmental Memorial, held on 10/23/13:
On Paul and Felice Bateman
At Sinai Temple, 10/19/16:
On Joe Rotman
Notices of the AMS Book review
"From Zero To Infinity, 50th Anniversary Edition"
by Constance Reid
When it comes to advice, I'm full of it
"Teaching Philosophy"
"Chalking It Up: Advice to a new TA"
"Resources for research"
"One introduction to mathematical research",
or "You've proved a theorem, now what?"
"Some thoughts on writing for the Putnam"

Areas of Specialization
Combinatorial methods in algebra, analysis and number theory,
usually involving polynomials or polytopes.
.pdf files of recent papers (most recent at top)
Preprint
Cantor set arithmetic (with Jayadev S. Athreya and Jeremy T. Tyson)
Preprint
On generic and maximal k-ranks of binary forms (with S. Lundqvist, A. Oneto and B. Shapiro)
Reprint
Binary forms with three different relative ranks (with Neriman Tokcan)
Preprint
The analogue of Hilbert's 1888 theorem for even symmetric
forms (with Charu Goel and Salma Kuhlmann)
Reprint
On the Choi-Lam analogue of Hilbert's 1888 theorem for
symmetric forms (with Charu Goel and Salma Kuhlmann)
Reprint
What does ``>'' really mean?
Preprint
Some new canonical forms for polynomials ,
Revised version
Preprint
Congruence properties of binary partition functions (with
Katherine Anders, Melissa Dennison and Jennifer Weber
Lansing) , Revised version
Preprint
On the sums of two cubes (with Jeremy
Rouse) , Revised version
Preprint
Blenders , Revised version
Preprint
On the length of binary forms , Final Revised version
Preprint
Polya's theorem with zeros (with Vicki
Powers and Mari Castle)
Reprint
Laws of inertia in higher degree binary forms
Preprint
On Hilbert's construction of positive polynomials
Preprint
Regularity properties of the Stern enumeration of the
rationals
Preprint
A quantitative Polya's theorem with zeros (with Vicki
Powers and Mari Castle)
Preprint
Clean lattice tetrahedra
Preprint
A quantitative Polya's theorem with corner zeros (with
Vicki Powers)
Preprint
Almost alternating sums (with Kevin O'Bryant and
Monika Serbinowska)
Preprint
Polynomials positive on unbounded rectangles (with Vicki
Powers)
Preprint
On the absence of uniform denominators in
Hilbert's 17th problem
Preprint
A new approach to Hilbert's theorem on ternary quartics
(with Vicki Powers, Claus Scheiderer and Frank Sottile)
Preprint
On the Diophantine equation x^{y} =
y^{mx} (with Michael Bennett)
Preprint
Patterns of dependence among powers of polynomials
Reprint
Lattice polytopes with distinct pair-sums (with M. D. Choi and
T. Y. Lam)
Preprint
A new bound for Polya's Theorem with applications to
polynomials positive on polyhedra (with Vicki Powers )
Preprint
Notes towards a constructive proof of Hilbert's Theorem on ternary
quartics (with Vicki Powers)
Preprint
Polynomials that are positive on an interval (with Vicki Powers)
Reprint
Some Concrete Aspects of Hilbert's 17th
Problem
Preprint
On the non-monotonicity of (|Im(z^{n} )|)
Preprint
The maximum angular gap among rectangular grid points (with
Zoltan Furedi)
Reprint
Homogeneous polynomial solutions to constant coefficient
PDE's over fields
Reprint
Uniform denominators in Hilbert's Seventeenth Problem
Preprint Sums of even
powers of real linear forms, chapter by chapter.
Reprint
An inequality for products of polynomials
Seminar Notes
Notes
Equal sums of higher powers of quadratic forms
Ph.D. Students
Prof. Julie Simon (Ph.D. 1989)
Prof. William Harris (Ph.D. 1992)
Ms. Colleen Kilker (1977-2002)
Prof. Ricardo Rojas (Ph.D. 2008)
Prof. Han Duong (Ph.D. 2008)
Prof. Supawadee Prugsapitak (Ph.D. 2009)
Prof. Melissa Dennison (Ph.D. 2010)
Dr. Milos Curcic (Ph.D. 2013)
Dr. Jennifer Weber Lansing (Ph.D. 2014)
Prof. Katie Anders (Ph.D. 2014)
Prof. Wipawee Tangjai (Ph.D. 2014)
Prof. Neriman Tokcan (Ph.D. 2017)
Mr. Sakulbuth Ekvittayaniphon
Mr. Benjamin Wright
Ms. Simone Sisneros-Thiry

Podcast!
Strongly Connected
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Updated January 17, 2018