Professor, Department of Mathematics
University of Illinois at Urbana-Champaign
- A uniqueness criterion for fractional iteration,
Ann. Polon. Math., XXX (1974), 219-224 (MR 50.2731).
- When is the iterate of a formal power series odd?,
J. Austral. Math. Soc., (Ser. A.) 28 (1979), 62-66 (MR
80g.39005).
- Banach spaces which satisfy linear identities, Pacific
J. Math., 74 (1978), 221-233 (MR 58.7045).
- Banach spaces with polynomial norms, Pacific J. Math.,
82 (1979), 223-235 (MR 83c.46007).
- Extremal psd forms with few terms, Duke Math. J., 45
(1978), 363-374 (MR 58.511).
- (With M. D. Choi and T. Y. Lam) Real zeros of positive
semidefinite forms, I, Math. Z., 171 (1980), 1-25 (MR
81d.10012).
- (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).
- (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).
- Some inequalities for products of power sums, Pacific
J. Math., 104 (1983), 443-463 (MR 84g.26015).
- (With S. Fitzpatrick) Skewness in Banach spaces,
Trans. Amer. Math. Soc., 5 (1983), 587-598 (MR
84a.46029).
- (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).
- (With J. Gillis and D. Zeilberger) On elementary methods in
positivity theory, SIAM J. Math. Anal., 14 (1983),
396-398 (MR 84i.42017).
- (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).
- Lattice point simplices, Discrete Math., 60 (1986),
219-242 (MR 87i.52022).
- Continued fractions and an annelidic pde,
Math. Intelligencer, 5 (1983), 61-63 (MR 86e.11007).
- Some extremal problems for continued fractions, Illinois
J. Math., 29 (1985), 261-279 (MR 87f.11007).
- (With P. Tiwari and D. West) Decomposition of product graphs into
complete bipartite subgraphs, Discrete Math., 57 (1985),
189-193 (MR 87i.05157).
- (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).
- Digital representations using the greatest integer function,
Trans. Amer. Math. Soc., 312 (1989), 355-375 (MR
89g.11010).
- (With M. D. Choi and T. Y. Lam) Even symmetric sextics,
Math. Z., 195 (1987), 559-580 (MR 88j.11019).
- A quantitative version of Hurwitz' theorem on the
arithmetic-geometric inequality, J. Reine Angew. Math.,
377 (1987), 108-112 (MR 88e.11019).
- (With T. Kratzke and D. West) Eigensharp graphs: decomposition
into complete bipartite subgraphs, Trans. Amer. Math. Soc.,
308 (1988), 637-653 (MR 89a.05104).
- Forms derived from the arithmetic-geometric inequality,
Math. Ann., 283 (1989), 431-464 (MR 90i.11043).
- (With M. D. Choi and T. Y. Lam) Positive sextics and Schur's
inequalities, J. Algebra, 141 (1991), 36-77 (MR
92i.11045).
- 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).
- 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).
- Sums of even powers of real linear forms,
Mem. Amer. Math. Soc., Volume 96, Number 463, March, 1992
(MR 93h.11043).
- An inequality for products of polynomials,
Proc. Amer. Math. Soc., 117 (1993), 1063-1073, (MR
93e.11058).
- (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).
- Uniform denominators in Hilbert's Seventeenth Problem,
Math. Z., 220 (1995), 75-98 (MR 96e:11056).
- Some constructions of spherical 5-designs, Linear Algebra
Appl., 226-228 (1995), 163-196 (MR 96f: 05048).
- (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).
- Homogeneous polynomial solutions to constant coefficient PDE's
over fields, Adv. Math., 117 (1996), 179-192 (MR
97a:12006).
- 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).
- On the non-monotonicity of (|Im(zn)|), J. Number Theory,
78 (1999), 144-148 (MR 2001a:11134).
- (With Z. Füredi) The maximal angular gap among rectangular grid
points, Per. Math. Hung., 36 (1998), 119-137 (MR 2000j:52024).
- (With V. Powers) Polynomials that are positive on an interval,
Trans. Amer. Math. Soc., 352 (2000), 4677-4692 (MR
2001b:12002).
- (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).
- (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).
- (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/0011068).
- 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/0106060).
- (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/0209072), Chinese translation in Math. Adv. Transl. .
- (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/0405475).
- On the absence of uniform denominators in Hilbert's Seventeenth
Problem, Proc. Amer. Math. Soc., 133 (2005), 2829-2834.
(MR2006d:12001, arXiv -- math/0306163).
- (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).
- (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).
- (With K. O'Bryant and M. Serbinowska) Almost alternating sums,
Amer. Math. Monthly, 113 (2006), 673--688
(MR 2007e:40001, arXiv -- math/0308087).
- (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).
- (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 (2008),179--187
(MR2405772, 2009i:52008).
- (With M. Castle and V. Powers) A quantitative Polya's Theorem
with zeros, J. Symbolic. Comp. 44 (2009), 1285--1290,
(MR2532172, 2010d:13029).
- Regularity properties of the Stern enumeration of the rationals,
Journal of Integer Sequences 11 (2008), Article
08.4.1, 17 pp., (MR2447843, 2009g:11016, arXiv -- math/0610601).
- Clean lattice tetrahedra, submitted for publication, arXiv --
math/0606227.
- On Hilbert's construction of positive polynomials,
submitted for publication, arXiv:0707.2156.
- Laws of inertia in higher degree binary forms,
Proc. of the AMS, 138 (2010),
815-826, (MR2566547, 2011e:11074. arXiv:0906.5559).
- (With M. Castle and V. Powers) Polya's Theorem with zeros,
J. Symbolic. Comp. 46 (2011), 1039--1048, (MR2819326,
2012g:13044).
- 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, 207-232, (MR3156559, arXiv:1007.5485).
- Blenders, in Notions of Positivity and the Geometry
of Polynomials (P. Branden, M.
Passare, M. Putinar, editors), Trends in Math., Birkhäuser,
Basel, 2011, 345-373, (MR3051175, arXiv1008.4533).
- (with J. Rouse) On the sums of two cubes,
Int. J. Number Theory, 7 (2011), 1863-1882,
(MR2854220, arXiv:1012.5801).
- (with K. Anders, M. Dennison, and J. W. Lansing) Congruence properties
of binary partition
functions, Ann. Combin., 17 (2013),
15-26, (MR3027571, arXiv:1102.5355).
- Some new canonical forms for polynomials,
Pac. J. Math., 266 (2013), 185-220,
(MR3105781, arXiv:1203.5722).
- (with C. Goel and S. Kuhlmann) On the Choi-Lam analogue of Hilbert's 1888
Theorem for Symmetric
Forms
Linear Algebra Appl., 496 (2016), 114-120. (MR 3464065, arXiv:1505.08145).
- (with C. Goel and S. Kuhlmann) The analogue of Hilbert's 1888
Theorem for Even Symmetric
Forms,
J. Pure Applied Algebra, 221 (2017), 1438-1448 (MR3599440, arXiv:1509.07482).
- (With N. Tokcan) Binary forms with three different relative
ranks,
Proc. Amer. Math. Soc., 145 (2017), 5169-5177 (MR3717946, arXiv:1608.08560).
- (with J. Athreya and J. Tyson) Cantor set arithmetic, American Mathematical Monthly,
126 (2019), 4-17
(MR3904605, arXiv:1711.08791).
- (with S. Lundqvist, A. Oneto, and B. Shapiro) On generic and
maximal k-ranks of binary forms,
J. Pure Applied Algebra, 223
(2019), 2062-2079 (MR3906540, arXiv:1711.05014).
- Linear dependent powers of binary quadratic forms,
Pac. J. Math. 303 (2019), 729-755, (MR4059953, arXiv:1903.11569).
- (with V. Powers) A note on mediated simplices,
J. Pure Applied Algebra 225, 2021 Paper No. 106608, 7 pp
(MR 4217929, arXiv:1909.11008).
- (with L. Matusevich) Sparse versions of the Cayley-Bacharach Theorem,
Port. Math., 78(2021), 129-146, (MR4301803,arXiv:1903.00075).
- Equal sums of two cubes of quadratic forms, Int. J. Number Theory., 17 (2021)
761-786, (MR4254775,arXiv:2002.00888 )
- (with Y. Alfassi and D. Keren) The non-tightness of a convex relaxation to rotation recovery
Sensors, 21 (2021), 7358, 11pp.
- (with K. Anders, M. Dawsey, and S. Sisneros-Thiry) Quotients of sums of powers of three, in preparation.
- (with C. Goel) Sums of squares of k-term forms, in preparation.
- (with C. Goel) A short analyst's proof that the set of irreducible complex
forms is open, in preparation.
- (with J. Lansing) The 19th century roots of the Stern sequence,
in preparation.
- A Stern introduction to combinatorial theory, book in preparation.
- Every math major should know this amazing theorem, in preparation.
- (with J. Rouse) Viete's Bolero, in preparation.
- On the optimal placement of unreliable observers, in preparation.
Updated March 29, 2022