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@illinois.edu

Spring 23 teaching

  • Course Organization for Graduate Reading Course

    • Fall 22 teaching

  • Course Organization for Graduate Reading Course

    • Fall 20 teaching

  • Math 417, Fall 2020
  • Course Organization for Graduate Reading Course, pre-covid

    • Links relevant to Math 499 talk, 10/1/18

  • "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).

    • 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

    • AMS Sectional Meeting, El Paso, Sept. 18, 2022

  • Slides of talk "Equal sums of two cubes"
  • Recording of talk

    Ramanujan Lecture 2021, University of Mauritius

  • Slides of talk, October 28, 2021

    Oberwolfach MFO Workshop 2010 -- Real Algebraic Geometry With a View Toward Hyperbolic Programming and Free Probability

  • Extended Abstract -- Lattice points in dilated simplices and sums of squares" March, 2020

    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 Algebraic 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)

  • Reprint The non-tightness of a convex relaxation to rotation recovery (With Y. Alfassi and D. Keren)
  • Preprint Equal sums of two cubes of quadratic forms
  • Reprint Sparse versions of the Cayley-Bacharach Theorem (with Laura Matusevich),
  • Reprint A note on mediated simplices (with Victoria Powers),
  • Preprint Linearly dependent powers of binary quadratic forms, Reprint
  • Preprint Sparse versions of the Cayley-Bacharach Theorem (with Laura Felicia Matusevich) Revised version
  • Reprint 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), Revised version
  • 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 xy = ymx (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(zn)|)
  • 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)
    Dr. 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)
    Prof. Sakulbuth Ekvittayaniphon (Ph.D. 2018)
    Prof. Simone Sisneros-Thiry (Ph.D. 2020)
    Prof. Dana Neidinger Neidmann (Ph.D. 2022)
    Prof. Grace Jaffe (Ph.D. 2023)
    Benjamin Wright

    Podcast!


    Strongly Connected Components

    Updated February 1, 2023

    Photo of Bern from Summer 2019: