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doc ///
Node
Key
exponents
(exponents, MonoidElement)
(exponents, RingElement)
Headline
the exponents of a polynomial
Usage
exponents m
Inputs
m:{RingElement,MonoidElement}
a monomial
Outputs
:List
for a monomial, provides the list of exponents;
for a polynomial, provides a list whose elements are the lists of exponents of the terms of @TT "f"@
Description
Example
R = ZZ/101[x,y,z];
exponents (f = x^2 - 7 + x*y*z^11 + y)
leadMonomial f
exponents leadMonomial f
SourceCode
(exponents, RingElement)
SeeAlso
content
coefficients
listForm
standardForm
monomials
Node
Key
content
(content, RingElement)
(content, RingElement, RingElement)
Headline
the content of a polynomial
Usage
content f
content(f, x)
Inputs
f:RingElement
x:RingElement
a variable in the ring of @TT "f"@
Outputs
:Ideal
the content of a matrix or polynomial
Description
Text
The content is the ideal in the base ring generated by the coefficients.
Example
R = ZZ[x,y]
content(4*x + 6*x^5)
content(4*x + 6*x^5, x)
content(4*x + 6*x^5, y)
generator oo
SourceCode
(content, RingElement)
(content, RingElement, RingElement)
SeeAlso
exponents
coefficients
listForm
standardForm
monomials
Node
Key
listForm
(listForm, RingElement)
(listForm, MonoidElement)
Headline
convert to list form
Usage
listForm f
Inputs
f:{RingElement,MonoidElement}
Outputs
:List
the polynomial or monomial in a form represented by nested lists.
Description
Text
A monomial is represented by the list of its exponents.
A polynomial is represented by lists of pairs @TT "(m, c)"@x, one for each term,
where @TT "m"@ is a list of exponents for monomial, and @TT "c"@ is the coefficient.
Example
R = ZZ/101[x,y,z];
listForm (x^2 - 7 + x*y*z^11 + y)
SourceCode
(listForm, RingElement)
SeeAlso
standardForm
content
coefficients
exponents
monomials
Node
Key
standardForm
(standardForm, RingElement)
(standardForm, MonoidElement)
Headline
convert to standard form
Usage
standardForm f
Inputs
Outputs
:HashTable
the polynomial or monomial in a form involving hash tables.
Description
Text
A polynomial is represented by hash tables in which the keys are hash tables representing the
monomials and the values are the coefficients. The monomials themselves are represented by hash tables
in which the keys are the variables and the values are the corresponding exponents.
Example
R = ZZ/101[x,y,z];
standardForm (x^2 - 7 + x*y*z^11 + y)
SourceCode
(standardForm, RingElement)
SeeAlso
listForm
content
coefficients
exponents
monomials
///
document {
Key => {coefficients,
(coefficients,Matrix),(coefficients,RingElement),
[coefficients,Monomials],[coefficients,Variables]},
Headline => "monomials and their coefficients",
Usage => "(M,C) = coefficients f",
Inputs => {
"f" => {"a one-row ", TO "Matrix", " with ", TT "n", " columns, say, or a ", TO "RingElement", ", to
be interpreted as a one-by-one matrix. (A future implementation will handle matrices with more than one row.)" },
Variables => List => {
"a list ", TT "v", " of variables.
If a value for this option is not specified, all of the (top-level) variables are used."
},
Monomials => {"a list or one-row matrix of monomials, each of which is formed using just variables in ", TT "v", "."}
},
Outputs => {
"M" => Matrix => {
"either the value of the ", TT "Monomials", " option, if specified (converted to a one-row matrix, if necessary), or
a one-row matrix of those monomials appearing in ", TT "f", " that involve just variables
in ", TT "v", ", in descending order. Let ", TT "m", " denote the number of columns it has."
},
"C" => Matrix => {
"the ", TT "m", " by ", TT "n", " matrix ", TT "C", " such that ", TT "C_(i,j)", " is
the coefficient in ", TT "f_(0,j)", " of the monomial ", TT "M_(0,i)", ".
In other words, ", TT "C", " is the unique matrix not involving the (specified) variables
such that ", TT "M*C == f", ", unless a value was specified for the ", TT "Monomials", " option that did not include
all the monomials in the variables ", TT "v", " used by ", TT "f"
}
},
EXAMPLE lines ///
R = QQ[a,b,c,d,e,f][x,y];
F = a*x^2+b*x*y+c*y^2
(M,C) = coefficients F
///,
"The resulting matrices have the following property.",
EXAMPLE lines ///
M*C === matrix F
///,
"The Sylvester matrix of two generic quadratic forms:",
EXAMPLE lines ///
G = d*x^2+e*x*y+f*y^2
P = matrix{{x*F,y*F,x*G,y*G}}
(M,C) = coefficients P
M*C === P
///,
"We may give the monomials directly. This is useful if we are taking coefficients
of several elements or matrices, and need a consistent choice of monomials.",
EXAMPLE lines ///
(M,C) = coefficients(P, Monomials=>{x^3,y^3,x^2*y,x*y^2})
///,
"If not all of the monomials are used, no error is signaled, but ", TT "M*C == P", " no longer holds.",
EXAMPLE lines ///
(M,C) = coefficients(P, Monomials=>{x^3,y^3})
M*C == P
///,
SeeAlso => {monomials,content,exponents,listForm,standardForm}
}
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