Lattice Polytopes with Distinct Pair-Sums |
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| Authors: | M. D. Choi T. Y. Lam B. Reznick |
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| Affiliation: | (1) Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3 choi@math.toronto.edu, CA;(2) Department of Mathematics, University of California, Berkeley, CA 94720, USA lam@math.berkeley.edu, US;(3) Department of Mathematics, University of Illinois, Urbana, IL 61801, USA reznick@math.uiuc.edu, US |
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| Abstract: | Let P be a lattice polytope in R n , and let P cap Z n = {v 1 ,ldots,v N } . If the N + Nchoose 2 points 2v 1 ,ldots, 2v N ;v 1 +v 2 ,ldots, v N-1 + v N are distinct, we say that P is a ``distinct pair-sum' or ``dps' polytope. We show that if P is a dps polytope in R n , then N≤ 2 n , and, for every n , we construct dps polytopes in R n which contain 2 n lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials. Received November 10, 2000, and in revised form June 28, 2001. Online publication November 2, 2001. |
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