This document is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. The copyright holder grants you permission to redistribute this document freely as a verbatim copy. Furthermore, the copyright holder permits you to develop any derived work from this document provided that the following conditions are met. a) The derived work acknowledges the fact that it is derived from this document, and maintains a prominent reference in the work to the original source. b) The fact that the derived work is not the original OpenMath document is stated prominently in the derived work. Moreover if both this document and the derived work are Content Dictionaries then the derived work must include a different CDName element, chosen so that it cannot be confused with any works adopted by the OpenMath Society. In particular, if there is a Content Dictionary Group whose name is, for example, `math' containing Content Dictionaries named `math1', `math2' etc., then you should not name a derived Content Dictionary `mathN' where N is an integer. However you are free to name it `private_mathN' or some such. This is because the names `mathN' may be used by the OpenMath Society for future extensions. c) The derived work is distributed under terms that allow the compilation of derived works, but keep paragraphs a) and b) intact. The simplest way to do this is to distribute the derived work under the OpenMath license, but this is not a requirement. If you have questions about this license please contact the OpenMath society at http://www.openmath.org. polyr http://www.openmath.org/cd http://www.openmath.org/cd/polyd.ocd 2006-03-30 2004-03-30 experimental 3 0 This CD contains operators to deal with polynomials and more precisely Recursive Polynomials. Note that polynomials are regarded as univariates in their most significant variable (as defined by the order in PolynomialRingR: the first variable to appear is the most significant), with monomials in decreasing order of exponent, and coefficients being polynomials in the rest of the variables. This means that polynomials have a unique representation, except for the fact that yz \in Z[x,y,z] could also be represented as x^0yz. This latter is discouraged, but currently not expressly forbidden. Original OpenMath v1.1 Poly 1997 Update to Current Format 1999-07-07 DPC Move the names of rings to setname1.ocd 1999-11-09 JHD Delete those items moved to the new poly.ocd 1999-11-14 JHD Convert to recursive polynomials 1999-11-20 JHD Definition of data-structure constructors The polynomial x^2*y^6 + 3*y^5 can be conceptually encoded as poly_r_rep(x, term(2,poly_r_rep(y, term(6,1))), term(0,poly_r_rep(y, term(5,3)))) It lies in polynomial_ring_r(Z,x,y) (and other rings, of course) The polynomial 2*y^3*z^5 + x + 1 can be conceptually encoded as poly_r_rep(x, term(1,1), term(0,poly_r_rep(y, term(3,poly_r_rep(z, term(5,2))), term(0,1)))) term application A constructor for monomials, that is products of powers and elements of the base ring. First argument is from N (the exponent of the variable implied by an outer poly_r_rep) second argument is a coefficient (from the ground field, or a polynomial in lesser variables). poly_r_rep application A constructor for the representation of polynomials. The first argument is the polynomial variable, the rest are monomials (in decreasing order of exponent). The polynomial x^2*y^6 + 3*x^0*y^5 = x^2*y^6 + 3*y^5 may be encoded as: 2 6 1 0 5 3 polynomial_r application The constructor of Recursive Polynomials. The first argument is the polynomial ring containing the polynomial and the second is a "poly_r_rep". The polynomial x^2*y^6 + 3*x^0*y^5 = x^2*y^6 + 3*y^5 in the polynomial ring with the integers as the coefficient ring and variables x,y in that order may be encoded as: 2 6 1 0 5 3 Polynomial ring constructor polynomial_ring_r application The constructor of a recursive polynomial ring. The first argument is a ring (the ring of the coefficients), the rest are the variables (in order).