Hilbert's third problem

Research interests: Approximation theory, numerical analysis, potential theory, polynomial inequalities
Erdõs Number: 2 (in 3 different ways)
Wiles Number in tennis:  2, because I played tennis with someone who played tennis with Andrew Wiles

Referee report on my paper about the Hilbert's 3rd problem:

"David Benko gives an elementary new resolution of Hilbert's third problem by resolving the gap in Bricard's 1896 approach. To fill a 110 year old gap, and to do it with a short and elementary argument is an excellent contribution, worthy of a place in "Proofs from the Book" (which he cites), and certainly a paper that the Monthly will want to accept."

In connection to Hilbert's third problem you may want to read a related article in the Western Scholar semi-annual magazine (pdf file) by Tommy Newton. The article is in the Spring 2007 issue, published at WKU.

Gauss               Hilbert

Publications:

  • (with D. Biles, M. Robinson and J. Spraker) Nystrom methods and singular second order differential equations (submitted)

  • (with C. Ernst and D. Lanphier) Aymptotic bounds on the integrity of graphs and separator theorems for graphs (submitted)

  • (with A. Kroó) A Weierstrass-type theorem for homogeneous polynomials, Trans. Amer. Math. Soc. (to appear)

  • A new approach to Hilbert's third problem, Amer. Math. Monthly 114 (2007), no. 8, 665-676 --- PDF

  • (with S. B. Damelin and P. D. Dragnev) On the support of the equilibrium measure for arcs of the unit circle and

      for real intervals, Electron. Trans. Numer. Anal. 25 (2006), 27-40 --- PDF

  • The support of the equilibrium measure, Acta Sci. Math. (Szeged) 70 (2004), no. 1-2, 35-55 --- PDF

  • (with T. Erdélyi, and J. Szabados) The full Markov-Newman inequality for Müntz polynomials on positive intervals,
      Proc. Amer. Math. Soc. 131 (2003), no. 8, 2385-2391 --- PDF

  • (with T. Erdélyi) Markov inequality for polynomials of degree n with m distinct zeros, J. Approx. Theory 122
      (2003), no. 2, 241-248 --- PDF

  • Approximation by weighted polynomials, J. Approx. Theory 120 (2003), no. 1, 153-182 --- PDF

  • (with V. Totik) Sets with interior extremal points for the Markoff inequality, J. Approx. Theory 110 (2001),
      no. 2, 261-265 --- PDF

Other works (non-refereed):

• The equilibrium measure and the Saff conjecture, Ph.D. dissertation (2006), University of Szeged, Hungary

  • Approximation by weighted polynomials, Ph.D. dissertation (2001), University of South Florida

  • On Slowly Diverging Series, Polygon (1995), V.2, 89-100

  • On the Number of Legal Chess Positions, Alpha (1995), no.1, 10-11

  • Fast decreasing polynomials, Master thesis (1995), University of Szeged

  • On the generalization of the Fundamental Theorem of Algebra, University of Szeged, Research Competition (1995)

  • On Slowly Converging Series, Polygon (1994), IV.2, 95-108

Collaborators:

  • D. Biles

  • S. Damelin

  • P. Dragnev

  • T. Erdélyi

  • C. Ernst

  • A. Kroó

  • D. Lanphier

  • M. Robinson

  • J. Spraker

  • J. Szabados

  • V. Totik

Problems:

I have some challenging math problems which have been used in math contests. For example:

(a)     The following problem of mine was one of the problems at the prestigious Schweitzer competition. (This is an annual math competition in Hungary. Students get 10-12 hard problems to solve; they can take them home and they have 10 days for thinking. They can even use the library.)

Let C denote the set of all convergent series with strictly positive terms. Let D denote the set of all divergent series with strictly positive terms. Does a bijection between C and D exist which satisfies the following property? :

If a_n and b_n are two elements of C and A_n and B_n are the corresponding elements in D, then

a_n / b_n tends to zero if and only if A_n / B_n tends to infinity.

(b)     The following was given in the KÖMAL mathematics journal in the hard problems category:

Does a three variable real polynomial P(x,y,z) exist such that P(x,y,z) is positive if and only if we can construct a triangle from three line segments whose lengths are |x|, |y| and |z|?

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Quotes:

"Your theory is crazy, but it's not crazy enough to be true." (Niels Bohr)

"I am always doing that which I can not do, in order that I may learn how to do it." (Pablo Picasso)

"If you have an apple and I have an apple and we exchange these apples then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas." (George Bernard Shaw)

"Relax, your solution is a few steps away..." (a slogan of Citibank)

   John von Neumann