# This is a file of SAGE commands to find the roots of a function that
# appears in Mas-Colell, Whinston, and Green, Microeconomic Theory,
# example 15.B.2, pp. 521. The code was proposed by William Stein
# in September 2007 and updated by Mike Hansen in April 2008 to work with
# SAGE version 2.11.  The code can be read and executed in a SAGE session
# by typing the following line:
# load "MWG15b2.sage"
#
# RDF() is shorthand for RealDoubleField(), a function that converts its
# argument into a real double-precision number.  For example, RDF(pi/e) 
# returns 1.15572734979 and RDF(8^(2/3)-8^(1/3)) returns 2.0.
# A polynomial ring is a set of polynomials with coefficients of a specified
# type -- e.g., integers, rationals, or double-precision reals.
# PolynomialRing() is a function that returns a polynomial ring with
# properties determined by its arguments. The variable or variables can be
# named and readied for use by listing them in diamond brackets following
# the name assigned to the ring.  For example, R.<y> = PolynomialRing(RDF)
# defines a univariate polynomial ring R in a variable y with real
# double-precision coefficients.  Similarly, P.<w,x> = PolynomialRing(QQ,2)
# defines a bivariate polynomial ring P in variables w and x with rational
# coefficients. If f is a univariate polynomial, f.roots() is a vector.
# If the elements of this vector are scalars, they are roots; if the elements
# of the vector are ordered pairs, the left member of a pair is a root and 
# the right member is its multiplicity.  To indicate which type of output is
# desired we can specify f.roots(multiplicities=False) or
# f.roots(multiplicities=True). The default option may vary from case to case.
# In our case, the default is True, which means that f.roots() is a vector of
# ordered pairs.  The left member of each pair is indexed as 0, the right
# member as 1.
#
# The function in example 15.B.2 is not a polynomial because it involves
# exponents that are not positive integers.  To convert it into a polynomial
# we replace x = p1/p2 with y = x^(1/9).  After finding the roots of the
# polynomial in y, we obtain the roots of the function in x by
# calculating x = y^9.
#
R.<y> = PolynomialRing(RDF)
# Let y be x^(1/9).
f = y + RDF(2^(8/9) - 2^(1/9))*(y^9-1) - y^8
v = f.roots(); v
print [a[0]^9 for a in v]