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-- -*- coding: utf-8 -*-
--- status:
--- author(s): L. Gold, Mahrud
--- notes:
doc ///
Node
Key
poincare
(poincare, Ring)
(poincare, Module)
(poincare, Ideal)
(poincare, MonomialIdeal)
installHilbertFunction
Headline
assemble degrees of a ring, module, or ideal into a polynomial
Usage
poincare M
Inputs
M:{Ring,Module,Ideal}
Outputs
:RingElement
in the @TO2 (degreesRing, "degrees ring")@ of the ambient ring, which is a Laurent polynomial ring whose
monomial @TT "T_0^(d_0) ... T_(r-1)^(d_(r-1))"@ correspond to the degree @TT "{d_0,...,d_(n-1)}"@
Description
Text
This function computes the numerator of the reduced @TO2 (hilbertSeries, "Hilbert series")@ of a
graded ring, or a graded module or ideal over a graded ring, which encodes information about the
twists in the minimal free resolution of the object.
Example
S = ZZ/101[w..z];
M = module monomialCurveIdeal(S, {1,3,4});
betti res M
poincare M
hilbertSeries M
Text
When $S$ is a $\ZZ^r$-graded ring in $n$ variables and $M$ a graded $S$-module, the Hilbert series
of $M$ is by definition the formal power series
$$ HS(T_0,\dots,T_{n-1}) = \sum_{d\in\ZZ^r} \mathrm{dim}(M_d) T_0^{d_0} \cdots T_{n-1}^{d_{n-1}}. $$
The coefficient of the term $T_0^{d_0} \cdots T_{n-1}^{d_{n-1}}$ in the series is the number of
basis elements of $M_{d_0,...,d_{n-1}}$. When the multi-degree has a single component, the coefficient
of $T^d$ is dimension of $M_d$.
Since the Hilbert series is additive in exact sequences, the function @TT "poincare"@ can be used
to get information about the minimal free resolution and hence the syzygies of $M$. For more details,
see @TO "Hilbert functions and free resolutions"@.
This polynomial is an element of the @TO2 (degreesRing, "degrees ring")@ of $S$. Notice that the
monomial ordering used in the degrees ring is @TO "RevLex"@, so the polynomials in it will be
displayed with the smallest exponents first.
Synopsis
Heading
compute the polynomial for a ring
Usage
poincare R
Description
Example
R = ZZ/101[x]/ideal(x^2);
poincare R
numerator hilbertSeries R
Text
Recall that the variables of the polynomial are the variables of the degrees ring.
Example
R = ZZ/101[x,y, DegreeRank => 2]/ideal(x^2*y);
poincare R
numerator hilbertSeries R
Synopsis
Heading
compute the polynomial for a module
Usage
poincare M
Description
Example
R = ZZ/101[w..z];
M = module monomialCurveIdeal(R, {1,3,4});
poincare M
numerator reduceHilbert hilbertSeries M
Synopsis
Heading
compute the polynomial for the quotient of a ring by an ideal
Usage
poincare I
Description
Text
The this command computes the numerator of the Hilbert series of the @TO comodule@ @TT "R^1/I"@,
where @TT "R"@ is the ring of @TT "I"@.
Example
R = ZZ/101[w..z];
I = monomialCurveIdeal(R, {1,3,4});
poincare I
poincare comodule I
numerator hilbertSeries I
numerator reduceHilbert hilbertSeries I
poincare module I
Description
Text
@HEADER2 "Caching of the result of poincare"@
When @TT "poincare M"@ is called on a module $M$, the result is cached in @TT "M.cache.poincare"@
for future reference. For ideals, the polynomial is cached in the @TO comodule@ of the ideal
and for matrices, it is cached in the @TO cokernel@ of the matrix.
This can speed the computation of Gröbner bases. For details, see @TO "computing Groebner bases"@.
If the numerator $p$ of the Hilbert series for a module $M$ is known by other means, it can be
manually stored by running:
Code
EXAMPLE { PRE //// poincare M = p -- cache poincare //// }
Text
This functionality used to be available by running @TT "installHilbertFunction(M, p)"@.
For example, in the following example, the Hilbert function of 3 random polynomials should be the
same as the Hilbert function for a complete intersection.
Example
R = ZZ/101[a..g];
I = ideal random(R^1, R^{3:-3});
p = poincare ideal(a^3, b^3, c^3)
poincare I = p
gbTrace = 3
time poincare I
time gens gb I;
Text
In this case, the savings is minimal, but often it can be dramatic.
Another important situation is to compute a Gröbner basis using a different monomial order.
Example
R = QQ[a..d];
I = ideal random(R^1, R^{3:-3});
time p = poincare I
S = QQ[a..d, MonomialOrder => Eliminate 2]
J = substitute(I, S)
poincare J = p
gbTrace = 3
time gens gb J;
selectInSubring(1, gens gb J)
Caveat
This function may behave unpredictably in presence of variables of degree zero.
SeeAlso
poincareN
(poincare, ChainComplex)
degreesRing
hilbertFunction
hilbertSeries
hilbertPolynomial
reduceHilbert
selectInSubring
"Elimination::Elimination"
///
-- This will be consumed into the Complexes package
doc ///
Node
Key
(poincare, ChainComplex)
Headline
assemble degrees of a chain complex into a polynomial
Usage
poincare C
Inputs
C:ChainComplex
Outputs
:RingElement
in the Laurent polynomial ring @TO2 (degreesRing, "degrees ring")@, whose variables correspond to the degrees of the ambient ring
Description
Text
We compute @TO poincare@ for a chain complex.
Example
R = ZZ/32003[a..h];
C = res ideal(a*b, c*d, e*f)
poincare C
Text
Note that since the Hilbert series is additive in exact sequences, for a free resolution this only depends
on the Betti numbers of the resolution. For more details, see @TO "Hilbert functions and free resolutions"@.
Example
b = betti C
poincare b
SeeAlso
poincareN
degreesRing
hilbertFunction
hilbertSeries
hilbertPolynomial
reduceHilbert
(poincare, BettiTally)
Node
Key
poincareN
(poincareN, ChainComplex)
Headline
assemble degrees into polynomial
Usage
poincareN C
Inputs
C:ChainComplex
Outputs
:RingElement
in the Laurent polynomial ring @TO2 (degreesRing, "degrees ring")@, whose variables correspond to the degrees of the ambient ring
Description
Text
This function encodes information about the degrees of basis elements of a free chain complex in a
polynomial. The polynomial has terms $S^i T_0^{d_0} \cdots T_{n-1}^{d_{n-1}}$ in it for each basis
element of @TT "C_i"@ with multi-degree @TT "{d_0,...,d_(n-1)}"@.
Example
R = ZZ/101[a,b,c, Degrees=>{1,1,2}];
C = res cokernel vars R
betti C
p = poincareN C
Text
Setting the @TT "S"@ variable to -1 gives the polynomial calculated by @TO (poincare, ChainComplex)@.
Example
sub(p, {S => -1})
poincare C
Text
Conversely, setting it to 1 gives the same polynomial for the direct sum of components of the complex.
Example
sub(p, {S => 1})
poincare sum C
SeeAlso
poincare
degreesRing
hilbertFunction
hilbertSeries
hilbertPolynomial
reduceHilbert
(poincare, BettiTally)
(sum, ChainComplex)
///
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