Dear Forum and dear Alexander,
recently Marc Bogaerds asked how to get the constant term of a polynomial.
Instead of using ExtRepPolynomialRatFun directly it is thinkable to
provide additional functions like 'ConstantTerm' or
'CoefficientOfPolynomial'
Now you may ask why this is useful?
The core GAP polynomial manipulation function set is sufficient but very
basic. Extensive polynomial manipulation using ExtRepPolynomialRatFun
would be very tedious and computationally expensive
( e.g. coefficient access compexity) and I experienced this in a recent
join project to compute Hurwitz map approximations.
An example and working definition for 'ConstantTerm' and
'CoefficientOfPolynomial' is attached. In fact, I defined even more
useful functions as attached
(e.g. evaluating/differentiating a polynomial tensor,
coercing polynomials to different rings,
counting variables appeared in a polynomial,
a more handy format for factors of a polynomial in comparison to
'Factors', etc.)
Please let me know, if you would like to see some this functions in a
special package or in the GAP core
or whatever.
Best,
Jakob
DeclareOperation( "IsMonomial", [IsObject] );
DeclareAttribute("CoefficientsDictionaryOfPolynomial",IsPolynomial);
DeclareOperation("CoefficientOfPolynomial",[ IsPolynomial, IsPolynomial ] );
DeclareOperation("ConstantTerm",[ IsPolynomial ] );
InstallMethod( IsMonomial, "", [IsObject],
function (monomial)
local monomData;
if not IsPolynomial (monomial) then
return false;
fi;
monomData := ExtRepPolynomialRatFun(monomial);
if Size(monomData) <>2 or not IsOne(monomData[2]) then
return false;
fi;
return true;
end
);
InstallMethod( CoefficientsDictionaryOfPolynomial ,
"get coefficient dict of a polynomial ", [ IsPolynomial ],
function( polynomial )
local coeffData, pos, dict;
if HasCoefficientsDictionaryOfPolynomial(polynomial) then
dict := CoefficientsDictionaryOfPolynomial(polynomial);
else
dict := NewDictionary([1,2,3,4], true);
coeffData := ExtRepPolynomialRatFun(polynomial);
for pos in [1..Size(coeffData)/2] do
AddDictionary(dict,coeffData[pos*2-1], coeffData[pos*2] );
od;
fi;
return dict ;
end
);
InstallMethod( CoefficientOfPolynomial ,
"get coefficient for a given monomial of an polynomial ", [ IsPolynomial,
IsPolynomial ],
function( polynomial, monomial )
local monomData, dict, coefficient;
if not IsMonomial ( monomial ) then
Error( "CoefficientOfPolynomial: second parameter is not a monomial !"
);
fi;
monomData := ExtRepPolynomialRatFun(monomial);
dict := CoefficientsDictionaryOfPolynomial(polynomial);
coefficient := LookupDictionary(dict, monomData[1]);
if coefficient=fail then
return Zero( CoefficientsFamily(FamilyObj(polynomial)) ) ;
fi;
return coefficient;
end
);
InstallMethod( ConstantTerm ,
"get constant part of a polynomial ", [ IsPolynomial ],
function( polynomial )
local dict, coefficient;
dict := CoefficientsDictionaryOfPolynomial(polynomial);
coefficient := LookupDictionary(dict, []);
if coefficient=fail then
return Zero( CoefficientsFamily(FamilyObj(polynomial)) ) ;
fi;
return coefficient;
end
);
# Example:
rng := PolynomialRing( ZmodnZ(11) ,["x","y"] );
indeterminates := IndeterminatesOfPolynomialRing(rng);
x := indeterminates[1];
y := indeterminates[2];
polynomial := (x^4-4)^3*(4*y^2+2);
CoefficientOfPolynomial(polynomial, x^4*y^2);
ConstantTerm(polynomial);
polynomial := 0*x^0;
ConstantTerm(polynomial);
polynomial := x^0;
ConstantTerm(polynomial);
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