Research Publications of Bruce Reznick

Professor, Department of Mathematics
University of Illinois at Urbana-Champaign

  1. A uniqueness criterion for fractional iteration, Ann. Polon. Math., XXX (1974), 219-224 (MR 50.2731).
  2. When is the iterate of a formal power series odd?, J. Austral. Math. Soc., (Ser. A.) 28 (1979), 62-66 (MR 80g.39005).
  3. Banach spaces which satisfy linear identities, Pacific J. Math., 74 (1978), 221-233 (MR 58.7045).
  4. Banach spaces with polynomial norms, Pacific J. Math., 82 (1979), 223-235 (MR 83c.46007).
  5. Extremal psd forms with few terms, Duke Math. J., 45 (1978), 363-374 (MR 58.511).
  6. (With M. D. Choi and T. Y. Lam) Real zeros of positive semidefinite forms, I, Math. Z., 171 (1980), 1-25 (MR 81d.10012).
  7. (With M. D. Choi, T. Y. Lam and A. Rosenberg) Sums of squares in some integral domains, J. Algebra, 65 (1980), 234-256 (MR 81h.10028).
  8. (With R. J. McEliece and J. B. Shearer) A Turán inequality arising in information theory, SIAM J. Math. Anal., 12 (1981), 931-934 (MR 83f.33008).
  9. Some inequalities for products of power sums, Pacific J. Math., 104 (1983), 443-463 (MR 84g.26015).
  10. (With S. Fitzpatrick) Skewness in Banach spaces, Trans. Amer. Math. Soc., 5 (1983), 587-598 (MR 84a.46029).
  11. (With M. D. Choi, M. Knebusch and T. Y. Lam) Transversal zeros and positive semidefinite forms, in Géométrie Algébrique Réelle et Formes Quadratiques, Rennes 1981 (J.-L. Colliot-Thélène, M. Coste, L. Mahé, M. F. Roy eds.), Lecture Notes in Math. 959 (1982), 273-298 (MR. 84b.10027).
  12. (With J. Gillis and D. Zeilberger) On elementary methods in positivity theory, SIAM J. Math. Anal., 14 (1983), 396-398 (MR 84i.42017).
  13. (With M. D. Choi, Z. D. Dai and T. Y. Lam) The pythagoras number of some affine algebras and local algebras, J. Reine Angew. Math., 336 (1982), 45-82 (MR 84f.12012).
  14. Lattice point simplices, Discrete Math., 60 (1986), 219-242 (MR 87i.52022).
  15. Continued fractions and an annelidic pde, Math. Intelligencer, 5 (1983), 61-63. (MR 86e.11007).
  16. Some extremal problems for continued fractions, Illinois J. Math., 29 (1985), 261-279 (MR 87f.11007).
  17. (With P. Tiwari and D. West) Decomposition of product graphs into complete bipartite subgraphs, Discrete Math., 57 (1985), 189-193 (MR 87i.05157).
  18. (With P. Erdös, A. Hildebrand, A. Odlyzko and P. Pudaite) The asymptotic behavior of a family of sequences, Pacific J. Math., 126 (1987) 227-241 (MR 89c.11024).
  19. Digital representations using the greatest integer function, Trans. Amer. Math. Soc., 312 (1989), 355-375 (MR 89g.11010).
  20. (With M. D. Choi and T. Y. Lam) Even symmetric sextics, Math. Z., 195 (1987), 559-580 (MR 88j.11019).
  21. A quantitative version of Hurwitz' theorem on the arithmetic-geometric inequality, J. Reine Angew. Math., 377 (1987), 108-112 (MR 88e.11019).
  22. (With T. Kratzke and D. West) Eigensharp graphs: decomposition into complete bipartite subgraphs, Trans. Amer. Math. Soc., 308 (1988) 637-653 (MR 89a.05104).
  23. Forms derived from the arithmetic-geometric inequality, Math. Ann., 283 (1989), 431-464 (MR 90i.11043).
  24. (With M. D. Choi and T. Y. Lam) Positive sextics and Schur's inequalities, J. Algebra, 141 (1991), 36-77 (MR 92i.11045).
  25. The sum of the squares of the parts of a partition, and some related questions, J. Number Theory, 33 (1989), 199-208 (MR 91d.11127).
  26. Some binary partition functions, in Analytic Number Theory, Proceedings of a Conference in Honor of Paul T. Bateman (B.C. Berndt, H.G. Diamond, H. Halberstam, A. Hildebrand eds.), 451-477, Birkhäuser, Boston, 1990 (MR 91k.11092).
  27. Sums of even powers of real linear forms, Mem. Amer. Math. Soc., Volume 96, Number 463, March, 1992 (MR 93h.11043).
  28. An inequality for products of polynomials, Proc. Amer. Math. Soc., 117 (1993), 1063-1073, (MR 93e.11058).
  29. (With M. D. Choi and T. Y. Lam) Sums of squares of real polynomials. Proc. Sympos. Pure Math., 58.2 (1995), 103-126 (MR 96f:11058).
  30. Uniform denominators in Hilbert's Seventeenth Problem, Math. Z., 220 (1995), 75-98 (MR 96e:11056).
  31. Some constructions of spherical 5-designs, Linear Algebra Appl., 226-228 (1995), 163-196 (MR 96f: 05048).
  32. (with M.D. Choi, T.Y. Lam, and A. Prestel) Sums of 2m-th powers of rational functions in one variable over real closed fields, Math. Z., 221 (1996), 93-112 (MR 96k:12003).
  33. Homogeneous polynomial solutions to constant coefficient PDE's over fields, Adv. Math., 117 (1996), 179-192 (MR 97a:12006).
  34. Some concrete aspects of Hilbert's 17th problem, Seminaire de Structures Algébriques Ordonnées, (F. Delon, M.A. Dickmann, D. Gondard eds), Publ. Équipe de Logique, Univ. Paris VII, Jan. 1996; revised version in Real Algebraic Geometry and Ordered Structures, (C. N. Delzell, J.J. Madden eds.) Cont. Math., 253 (2000), 251-272 (MR 2001i:11042).
  35. On the non-monotonicity of (|Im(zn)|), J. Number Theory, 78 (1999), 144-148 (MR 2001a:11134).
  36. (With Z. Füredi) The maximal angular gap among rectangular grid points, Per. Math. Hung., 36 (1998), 119-137 (MR 2000j:52024).
  37. (With V. Powers) Polynomials that are positive on an interval, Trans. Amer. Math. Soc., 352 (2000), 4677-4692 (MR 2001b:12002).
  38. (With V. Powers) Notes towards a constructive proof of Hilbert's Theorem on ternary quartics, Proceedings, Quadratic forms and their applications, Dublin 1999 (A. Ranicki ed.) Cont. Math. , 272 (2000), 209-227 (MR 2001h:11049).
  39. (With V. Powers) A new bound for Pòlya's Theorem with applications to polynomials positive on polyhedra, J. Pure Appl. Alg., 164 (2001), 221-229 (MR 2002g:14087).
  40. (With M. D. Choi and T. Y. Lam) Lattice polytopes with distinct pair-sums, Discrete Comput. Geom., 27 (2002), 65-72 (MR 2003e:52019, arXiv -- math.CO/0011068).
  41. Patterns of dependence among powers of polynomials, Algorithmic and Quantitative Real Algebraic Geometry (S. Basu, L. Gonzalez-Vega, eds.), DIMACS Ser. in Discrete Mathematics and Theoretical Computer Science, 60 (2003), 101-121 (MR 2004e:11036, arXiv -- math.AG/0106060).
  42. (With M. Bennett) Positive rational solutions to xy = ymx: a number-theoretic excursion, Amer. Math. Monthly , 111 (2004), 13-21 (MR 2004j:11033, arXiv -- math.NT/0209072), Chinese translation in Math. Adv. Transl. .
  43. (With V. Powers, C. Scheiderer and F. Sottile) A new approach to Hilbert's theorem on ternary quartics, C. R. Math. Acad Sci. Paris, 339 (2004), 617 - 620, (MR2005i:11051, arXiv -- math.AG/0405475).
  44. On the absence of uniform denominators in Hilbert's Seventeenth Problem, Proc. Amer. Math. Soc., 133 (2005), 2829-2834. (MR2006d:12001, arXiv -- math.AG/0306163).
  45. (With V. Powers) Polynomials positive on unbounded rectangles, Positive polynomials in Control (A. Garulli, D. Henrion eds.), Lecture Notes in Control and Inform. Sci. , 312 , 151-163, Springer, Berlin 2005 (MR 2005m:14108).
  46. (With M. Beck, B. Chen, L. Fukshansky, C. Haase, A. Knutson, S. Robins, A. Schürmann) Problems from the Cottonwood Room, Integer Points in Polyhedra -- Geometry, Number Theory, Algebra, Optimization (A. Barvinok, M. Beck, C. Haase, B. Reznick, V. Welker eds.) Cont. Math., 374 (2005), 179-191 (MR 2134767).
  47. (With K. O'Bryant and M. Serbinowska) Almost alternating sums, Amer. Math. Monthly , 113 (2006), 673--688 (MR 2007e:40001, arXiv -- math.NT/0308087).
  48. (With V. Powers) A quantitative Polya's Theorem with corner zeros, In Proceedings of the 2006 International Symposium of Symbolic and Algebraic Computation, J.-G. Dumas editor, New York, ACM Press, 285 -- 290, (MR2289132).
  49. (With M. Beck, B. Nill, C. Savage, I. Soprunov, Z. Xu) Let me tell you my favorite lattice point problem..., Integer Points in Polyhedra--Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, (M. Beck, C. Haase, B. Reznick, M. Vergne, V. Welker, R. Yoshida eds.) Cont. Math. 452, pp. 179--187, (2008), MR2405772 (2009i:52008).
  50. (With M. Castle and V. Powers) A quantitative Polya's Theorem with zeros, J. Symbolic. Comp. 44 (2009), 1285--1290, MR2532172 (2010d:13029).
  51. Regularity properties of the Stern enumeration of the rationals, Journal of Integer Sequences 11 (2008), Article 08.4.1, 17 pp., arXiv -- math.NT/0610601, MR2447843 (2009g:11016).
  52. Clean lattice tetrahedra, submitted for publication, arXiv -- math.CO/0606227.
  53. On Hilbert's construction of positive polynomials, submitted for publication, arXiv:0707.2156.
  54. Laws of inertia in higher degree binary forms, Proc. of the AMS, 138 (2010), 815-826, http://arxiv.org/pdf/0906.5559.pdf, MR2566547 (2011e:11074).
  55. (With M. Castle and V. Powers) Polya's Theorem with zeros, J. Symbolic. Comp. 46 (2011), 1039--1048, MR2819326 (2012g:13044).
  56. On the length of binary forms, Quadratic and Higher Degree Forms, (K. Alladi, M. Bhargava, D. Savitt, P. Tiep, eds.), Developments in Math. 31 (2013), Springer, New York, pp. 207-232, http://arxiv.org/pdf/1007.5485.pdf, MR3156559.
  57. Blenders, in Notions of Positivity and the Geometry of Polynomials (P. Branden, M. Passare, M. Putinar, editors), Trends in Math., Birkhäuser, Basel, 2011, pp.345-373, http://arxiv.org/pdf/1008.4533v1, MR3051175
  58. (with J. Rouse) On the sums of two cubes, Int. J. Number Theory, 7 (2011), 1863-1882, http://arxiv.org/pdf/1012.5801, MR2854220.
  59. (with K. Anders, M. Dennison and J. W. Lansing) Congruence properties of binary partition functions, Ann. Combin., 17 (2013), pp. 15-26,http://front.math.ucdavis.edu/1102.5355, MR3027571.
  60. Some new canonical forms for polynomials, Pac. J. Math., 266 (2013), 185-220, http://arxiv.org/abs/1203.5722, MR3105781.
  61. (with C. Goel and S. Kuhlmann) On the Choi-Lam analogue of Hilbert's 1888 Theorem for Symmetric Forms, http://front.math.ucdavis.edu/1505.08145v2, Linear Algebra Appl., 496 (2016), 114-120. MR 3464065.
  62. (with C. Goel and S. Kuhlmann) The analogue of Hilbert's 1888 Theorem for Even Symmetric Forms, http://front.math.ucdavis.edu/1509.07482, J. Pure Applied Algebra, 221 (2017), 1438-1448. MR3599440
  63. (With N. Tokcan) Binary forms with three different relative ranks, http://front.math.ucdavis.edu/1608.08560, Proc. Amer. Math. Soc., 145 (2017), 5169-5177.
  64. (with J. Athreya and J. Tyson) Cantor set arithmetic, to appear in the American Mathematical Monthly, https://arxiv.org/abs/1711.08791.
  65. (with S. Lundqvist, A. Oneto and B. Shapiro) On generic and maximal k-ranks of binary forms, submitted for publication, http://front.math.ucdavis.edu/1711.05014
  66. (with C. Goel) Sums of squares of k-term forms, in manuscript
  67. (with L. Matusevich) Sparse versions of the Cayley-Bacharach Theorems, in preparation.
  68. (with G. Blekherman) Psd ternary forms with many zeros, in preparation.
  69. (with J. Lansing) The 19th century roots of the Stern sequence, in preparation.
  70. Linear dependence of powers of quadratic forms, in preparation.
  71. A Stern introduction to combinatorial theory, book in preparation.
  72. Every math major should know this crazy theorem, in preparation.
  73. (with J. Rouse) Viete's Bolero, in preparation.
  74. On the optimal placement of unreliable observers, in preparation.
Updated November 27, 2017