)abbrev category ACFS AlgebraicallyClosedFunctionSpace
++ Author: Manuel Bronstein
++ Date Created: 31 October 1988
++ Date Last Updated: 7 October 1991
++ Description:
++ Model for algebraically closed function spaces.
++ Keywords: algebraic, closure, field.
AlgebraicallyClosedFunctionSpace(R: IntegralDomain):
Category == Join(AlgebraicallyClosedField, FunctionSpace R) with
rootOf : $ -> $
++ rootOf(p) returns y such that \spad{p(y) = 0}.
++ Error: if p has more than one variable y.
rootsOf: $ -> List $
++ rootsOf(p, y) returns \spad{[y1,...,yn]} such that \spad{p(yi) = 0};
++ Note: the returned symbols y1,...,yn are bound in the interpreter
++ to respective root values.
++ Error: if p has more than one variable y.
rootOf : ($, Symbol) -> $
++ rootOf(p,y) returns y such that \spad{p(y) = 0}.
++ The object returned displays as \spad{'y}.
rootsOf: ($, Symbol) -> List $
++ rootsOf(p, y) returns \spad{[y1,...,yn]} such that \spad{p(yi) = 0};
++ The returned roots display as \spad{'y1},...,\spad{'yn}.
++ Note: the returned symbols y1,...,yn are bound in the interpreter
++ to respective root values.
zeroOf : $ -> $
++ zeroOf(p) returns y such that \spad{p(y) = 0}.
++ The value y is expressed in terms of radicals if possible,and otherwise
++ as an implicit algebraic quantity.
++ Error: if p has more than one variable.
zerosOf: $ -> List $
++ zerosOf(p) returns \spad{[y1,...,yn]} such that \spad{p(yi) = 0}.
++ The yi's are expressed in radicals if possible.
++ The returned symbols y1,...,yn are bound in the interpreter
++ to respective root values.
++ Error: if p has more than one variable.
zeroOf : ($, Symbol) -> $
++ zeroOf(p, y) returns y such that \spad{p(y) = 0}.
++ The value y is expressed in terms of radicals if possible,and otherwise
++ as an implicit algebraic quantity
++ which displays as \spad{'y}.
zerosOf: ($, Symbol) -> List $
++ zerosOf(p, y) returns \spad{[y1,...,yn]} such that \spad{p(yi) = 0}.
++ The yi's are expressed in radicals if possible, and otherwise
++ as implicit algebraic quantities
++ which display as \spad{'yi}.
++ The returned symbols y1,...,yn are bound in the interpreter
++ to respective root values.
add
rootOf(p:$) ==
empty?(l := variables p) => error "rootOf: constant expression"
rootOf(p, first l)
rootsOf(p:$) ==
empty?(l := variables p) => error "rootsOf: constant expression"
rootsOf(p, first l)
zeroOf(p:$) ==
empty?(l := variables p) => error "zeroOf: constant expression"
zeroOf(p, first l)
zerosOf(p:$) ==
empty?(l := variables p) => error "zerosOf: constant expression"
zerosOf(p, first l)
zeroOf(p:$, x:Symbol) ==
n := numer(f := univariate(p, kernel(x)$Kernel($)))
positive? degree denom f => error "zeroOf: variable appears in denom"
degree n = 0 => error "zeroOf: constant expression"
zeroOf(n, x)
rootOf(p:$, x:Symbol) ==
n := numer(f := univariate(p, kernel(x)$Kernel($)))
positive? degree denom f => error "roofOf: variable appears in denom"
degree n = 0 => error "rootOf: constant expression"
rootOf(n, x)
zerosOf(p:$, x:Symbol) ==
n := numer(f := univariate(p, kernel(x)$Kernel($)))
positive? degree denom f => error "zerosOf: variable appears in denom"
degree n = 0 => empty()
zerosOf(n, x)
rootsOf(p:$, x:Symbol) ==
n := numer(f := univariate(p, kernel(x)$Kernel($)))
positive? degree denom f => error "roofsOf: variable appears in denom"
degree n = 0 => empty()
rootsOf(n, x)
rootsOf(p:SparseUnivariatePolynomial $, y:Symbol) ==
(r := retractIfCan(p)@Union($,"failed")) case $ => rootsOf(r::$,y)
rootsOf(p, y)$AlgebraicallyClosedField_&($)
zerosOf(p:SparseUnivariatePolynomial $, y:Symbol) ==
(r := retractIfCan(p)@Union($,"failed")) case $ => zerosOf(r::$,y)
zerosOf(p, y)$AlgebraicallyClosedField_&($)
zeroOf(p:SparseUnivariatePolynomial $, y:Symbol) ==
(r := retractIfCan(p)@Union($,"failed")) case $ => zeroOf(r::$, y)
zeroOf(p, y)$AlgebraicallyClosedField_&($)
