17,041 research outputs found
Using Representation Theorems for Proving Polynomials Non-negative
Proving polynomials non-negative when variables range on a
subset of numbers (e.g., [0, +∞)) is often required in many applications
(e.g., in the analysis of program termination). Several representations for
univariate polynomials P that are non-negative on [0, +∞) have been
investigated. They can often be used to characterize the property, thus
providing a method for checking it by trying a match of P against the
representation. We introduce a new characterization based on viewing
polynomials P as vectors, and find the appropriate polynomial basis B
in which the non-negativeness of the coordinates [P]B representing P in
B witnesses that P is non-negative on [0, +∞). Matching a polynomial
against a representation provides a way to transform universal sentences
∀x ∈ [0, +∞) P(x) ≥ 0 into a constraint solving problem which can be
solved by using efficient methods. We consider different approaches to
solve both kind of problems and provide a quantitative evaluation of
performance that points to an early result by P´olya and Szeg¨o’s as an
appropriate basis for implementations in most cases.Lucas Alba, S. (2014). Using Representation Theorems for Proving Polynomials Non-negative. En Artificial Intelligence and Symbolic Computation: 12th International Conference, AISC 2014, Seville, Spain, December 11-13, 2014. Proceedings. Springer Verlag (Germany). 21-33. doi:10.1007/978-3-319-13770-4_4S2133Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving Termination Properties with mu-term. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 201–208. Springer, Heidelberg (2011)Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Berlin (2006)Bernstein, S.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Communic. Soc. Math. de Kharkow 13(2), 1–2 (1912)Bernstein, S.: Sur la répresentation des polynômes positifs. Communic. Soc. Math. de Kharkow 14(2), 227–228 (1915)Borralleras, C., Lucas, S., Oliveras, A., RodrÃguez, E., Rubio, A.: SAT Modulo Linear Arithmetic for Solving Polynomial Constraints. Journal of Automated Reasoning 48, 107–131 (2012)Boudaoud, F., Caruso, F., Roy, M.-F.: Certificates of Positivity in the Bernstein Basis. Discrete Computational Geometry 39, 639–655 (2008)Choi, M.D., Lam, T.Y., Reznick, B.: Sums of squares of real polynomials. In: Proc. of the Symposium on Pure Mathematics, vol. 4, pp. 103–126. American Mathematical Society (1995)Contejean, E., Marché, C., Tomás, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. Journal of Automated Reasoning 32(4), 315–355 (2006)Hilbert, D.: Ãœber die Darstellung definiter Formen als Summe von Formenquadraten. Mathematische Annalen 32, 342–350 (1888)Hong, H., JakuÅ¡, D.: Testing Positiveness of Polynomials. Journal of Automated Reasoning 21, 23–38 (1998)Karlin, S., Studden, W.J.: Tchebycheff systems: with applications in analysis and statistics. Interscience, New York (1966)Lucas, S.: Polynomials over the reals in proofs of termination: from theory to practice. RAIRO Theoretical Informatics and Applications 39(3), 547–586 (2005)Polya, G., Szegö, G.: Problems and Theorems in Analysis II. Springer (1976)Powers, V., Reznick, B.: Polynomials that are positive on an interval. Transactions of the AMS 352(10), 4677–4692 (2000)Powers, V., Wörmann, T.: An algorithm for sums of squares of real polynomials. Journal of Pure and Applied Algebra 127, 99–104 (1998
Paley-Wiener theorems for the Dunkl transform
We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl
transform and prove three instances thereof, one of which involves a limit
transition from Opdam's results for the graded Hecke algebra. Furthermore, the
connection between Dunkl operators and the Cartan motion group is established.
It is shown how the algebra of radial parts of invariant differential operators
can be described explicitly in terms of Dunkl operators, which implies that the
generalized Bessel functions coincide with the spherical functions. In this
context, the restriction of Dunkl's intertwining operator to the invariants can
be interpreted in terms of the Abel transform. Using shift operators we also
show that, for certain values of the multiplicities of the restricted roots,
the Abel transform is essentially inverted by a differential operator.Comment: LaTeX, 26 pages, no figures. References updated and minor changes,
mathematically identical to the first version. To appear in Trans. Amer.
Math. So
Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
The purpose of this note is to survey a methodology to solve systems of
polynomial equations and inequalities. The techniques we discuss use the
algebra of multivariate polynomials with coefficients over a field to create
large-scale linear algebra or semidefinite programming relaxations of many
kinds of feasibility or optimization questions. We are particularly interested
in problems arising in combinatorial optimization.Comment: 28 pages, survey pape
- …