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jets package updated to version 1.2
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galettof committed Oct 15, 2024
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265 changes: 248 additions & 17 deletions M2/Macaulay2/packages/Jets.m2
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Original file line number Diff line number Diff line change
@@ -1,5 +1,5 @@
--------------------------------------------------------------------------------
-- Copyright 2021-2022 Federico Galetto, Nicholas Iammarino
-- Copyright 2021-2024 Federico Galetto, Nicholas Iammarino
--
-- This program is free software: you can redistribute it and/or modify it under
-- the terms of the GNU General Public License as published by the Free Software
Expand All @@ -17,8 +17,8 @@

newPackage(
"Jets",
Version => "1.1",
Date => "June 10, 2022",
Version => "1.2",
Date => "October 15, 2024",
AuxiliaryFiles => true,
Authors => {
{
Expand Down Expand Up @@ -54,7 +54,7 @@ newPackage(



importFrom(MinimalPrimes, {"radical"});
importFrom(MinimalPrimes, {"radical","minimalPrimes"});

export {
"JJ",
Expand All @@ -69,7 +69,9 @@ export {
"jetsProjection",
"jetsInfo",
"principalComponent",
"Saturate"
"Saturate",
"liftingFunction",
"liftingMatrix"
}

jetsOptions = {
Expand Down Expand Up @@ -354,7 +356,7 @@ jets(ZZ,HyperGraph) := HyperGraph => o -> (n,G) -> (
hyperGraph E
)

jets(ZZ,AffineVariety) := o -> (n,V) -> (
jets(ZZ,AffineVariety) := Variety => o -> (n,V) -> (
if n<0 then error("jets order must be a non-negative integer");
R := ring V;
JR := jets(n,R,Projective=> o.Projective);
Expand Down Expand Up @@ -417,30 +419,83 @@ JJ = new ScriptedFunctor from {

--compute an ideal whose vanishing locus is the
--principal component of the jets of an ideal
--changed in v1.2 with a faster algorithm for monomial ideals
--and to fix the behavior for reducible varieties
-- FG's note: I tried an option for bypassing the computation
-- of minimal primes, but for some reason this method appears to
-- work faster if minimal primes are found first
-- (at least for 2x2 minors of a generic 3x3 matrix)
principalComponent = method(Options=>{Saturate=>true},TypicalValue=>Ideal)
principalComponent(ZZ,Ideal) := o -> (n,I) -> (
if n<0 then error("jets order must be a non-negative integer");
-- find minimal primes
mp := minimalPrimes I;
-- for a monomial ideal use shortcut from Galetto-Iammarino-Yu
if isMonomialIdeal(I) then (
return intersect(apply(mp, P -> jets(n,P)));
);
-- compute the singular locus of I by breaking up components
-- and finding singular locus of each
-- (this is necessary as of v1.24.05 because the singularLocus
-- method only works for irreducible ideals)
singComp := apply(mp, P -> ideal singularLocus P);
-- then also intersect components two at a time
pairwiseInt := apply(subsets(mp,2),sum);
-- and finally take the union
sing := intersect(singComp|pairwiseInt);
-- compute jets of I
JI := jets(n,I);
-- get the jets projection
R := ring I;
p := jetsProjection(n,0,R);
-- identify original ambient ring with 0-jets
i := map(source p,R,vars source p);
--compute the singular locus of I
--map it to the zero jets via the map i
--map the singular locus to the zero jets via the map i
--then to the n jets via the map p
sing := p(i(ideal singularLocus I));
sing0 := p i sing;
--default is to saturate JI wrt sing
if o.Saturate then (
saturate(JI,sing)
saturate(JI,sing0)
)
--if JI is radical, colon is enough
else (
JI:sing
JI:sing0
)
)

-- the following methods (liftingFunction, liftingMatrix)
-- follow the definitions in the paper by Galetto-Iammarino-Yu
-- unexported recursive computation of lifting function
lf = (s,j,k) -> (
-- deal with edge cases
if (k<j or k>(s+1)*j) then return 0_ZZ;
if (k==j) then return ((s+1)^j)_ZZ;
if (k==(s+1)*j) then return 1_ZZ;
-- recursive computation
sum(min(k,s+1), i -> binomial(s+1,i+1) * mlf(s,j-1,k-i-1) )
)

-- speeds up computation by storing values
mlf = memoize lf

-- lifting function method for user
liftingFunction = method(TypicalValue => ZZ);
liftingFunction(ZZ,ZZ,ZZ) := ZZ => (s,j,k) -> (
-- check arguments are nonnegative
if (s<0 or j<0 or k<0) then error("arguments should be nonnegative");
mlf(s,j,k)
)

-- enter values of lifting function in a matrix
-- row/column indices start at zero
liftingMatrix = method(TypicalValue => Matrix);
liftingMatrix(ZZ,ZZ,ZZ) := Matrix => (s,r,c) -> (
-- check arguments are nonnegative
if (s<0) then error("first argument should be nonnegative");
if (r<=0 or c<=0) then error("second and third argument should be positive");
matrix table(r,c, (j,k) -> mlf(s,j,k) )
)

beginDocumentation()
----------------------------------------------------------------------
-- TESTS
Expand Down Expand Up @@ -547,6 +602,22 @@ TEST ///
p = jetsProjection(3,1,R)
assert(ring p JI === jets(3,R))
///

-- for lifting function
TEST ///
M=matrix{{1,0,0,0,0,0,0,0,0},
{0,2,1,0,0,0,0,0,0},
{0,0,4,4,1,0,0,0,0},
{0,0,0,8,12,6,1,0,0},
{0,0,0,0,16,32,24,8,1}}
assert(liftingMatrix(1,5,9) === M)
N=matrix{{1,0,0,0,0,0,0,0,0,0,0,0,0},
{0,3,3,1,0,0,0,0,0,0,0,0,0},
{0,0,9,18,15,6,1,0,0,0,0,0,0},
{0,0,0,27,81,108,81,36,9,1,0,0,0},
{0,0,0,0,81,324,594,648,459,216,66,12,1}}
assert(liftingMatrix(2,5,13) === N)
///
----------------------------------------------------------------------
-- Documentation
----------------------------------------------------------------------
Expand All @@ -571,29 +642,42 @@ Node
the radical of jets of monomial ideals, a function
to construct jets of graphs, a method for principal components of jets,
and an option for jets with "projective" gradings.

@HEADER4 "Version history:"@

@UL {(BOLD "1.1: ", "JSAG version."),
(BOLD "1.2: ", "Improved method for principal components.
Added methods for invariants of principal components
of monomial ideals.")
}@
References
@arXiv("math/0612862","L. Ein and M. Mustaţă,
Jet schemes and singularities.")@
@arXiv("2104.08933","F. Galetto, E. Helmick, and M. Walsh,
Jet graphs.")@

@arXiv("math/0407113","P. Vojta,
Jets via Hasse-Schmidt Derivations.")@

@HREF("https://doi.org/10.1080/00927870500454927",
"R.A. Goward and K.E. Smith,
The jet scheme of a monomial scheme.")@

@arXiv("math/0407113","P. Vojta,
Jets via Hasse-Schmidt Derivations.")@

@arXiv("2104.08933","F. Galetto, E. Helmick, and M. Walsh,
Jet graphs.")@

@arXiv("2407.01836","F. Galetto, N. Iammarino, and T. Yu,
Jets and principal components of monomial ideals, and very well-covered graphs")@
Subnodes
:Package methods
jets
jetsProjection
jetsRadical
principalComponent
liftingFunction
:Examples from the literature
"Example 1"
"Example 2"
"Example 3"
"Example 4"
:Technical information
"Storing Computations"

Expand Down Expand Up @@ -1226,6 +1310,11 @@ Node
P = primaryDecomposition jets(2,I)
any(P,c -> c == PC)
PC == intersect(select(P,c -> degree c == 1))
Text
If $I$ is a monomial ideal, this method uses a different characterization
of the principal component (see Theorem 6.7 in
@arXiv("2407.01836","F. Galetto, N. Iammarino, and T. Yu,
Jets and principal components of monomial ideals, and very well-covered graphs")@).
Caveat
This function requires computation of a singular locus,
a saturation (or quotient), and jets, with each step being
Expand Down Expand Up @@ -1263,6 +1352,82 @@ Node
JJ_2 R
JJ_2 I

Node
Key
liftingFunction
(liftingFunction,ZZ,ZZ,ZZ)
Headline
compute values of a lifting function
Usage
liftingFunction(s,j,k)
Inputs
s:ZZ
a natural number
j:ZZ
a natural number
k:ZZ
a natural number
Outputs
:ZZ
the number of cardinality $k$ lifts of a cardinality $j$ set under depolarization
Description
Text
This function was added in version 1.2 of the package @TO Jets@.

Given a set $X$ and a natural number $s$, let $\mathcal{J}_s (X)$
be the set that contains the elements $x_0,\dots,x_s$ for every
element $x\in X$. The @EM "depolarization"@ map
$\delta_s \colon \mathcal{J}_s (X)\to X$ is defined by
$\delta_s (x_i) = x$ for every $x\in X$ and $i\in \{0,\dots,s\}$.

The @EM "lifting function"@ $l_s (j,k)$ counts the number
of subsets $V\subseteq \mathcal{J}_s (X)$ of cardinality $k$
such that $\delta_s (V) = U$, where $U\subseteq X$ is a fixed
subset of cardinality $j$. Note that this number does not
depend on $U$ but only on its cardinality.
See @arXiv("2407.01836","F. Galetto, N. Iammarino, and T. Yu,
Jets and principal components of monomial ideals, and very well-covered graphs")@
for computing this function.
Example
liftingFunction(1,2,3)
liftingFunction(2,2,3)
liftingFunction(1,3,2)
liftingFunction(1,0,0)
Text
For uses of the lifting function, see @TO "Example 4"@.
Subnodes
liftingMatrix

Node
Key
liftingMatrix
(liftingMatrix,ZZ,ZZ,ZZ)
Headline
arrange values of lifting function in a matrix
Usage
liftingMatrix(s,r,c)
Inputs
s:ZZ
a natural number
r:ZZ
a positive integer
c:ZZ
a positive integer
Outputs
:Matrix
@TT "r"@ by @TT "c"@, whose entries are the values of the order @TT "s"@ lifting function
Description
Text
This function was added in version 1.2 of the package @TO Jets@.

This function collects the values of the @TO "liftingFunction"@
$l_s (j,k)$ and arranges them in an @TT "r"@ by @TT "c"@ matrix $L_s (j,k)$
with row index $j\geqslant 0$ and column index $k\geqslant 0$.
Example
liftingMatrix(2,3,5)
Text
For uses of the lifting matrix, see @TO "Example 4"@.

Node
Key
"Example 1"
Expand Down Expand Up @@ -1432,5 +1597,71 @@ Node
Example
apply({P_0,I2}, X -> hilbertSeries(X,Reduce=>true))
numerator (first oo) == (numerator last oo)^2

Node
Key
"Example 4"
Headline
invariants of principal jets of monomial ideals
Description
Text
This follows Examples 7.5 and 7.7 in
@arXiv("2407.01836","F. Galetto, N. Iammarino, and T. Yu,
Jets and principal components of monomial ideals, and very well-covered graphs")@.

Consider the following squarefree monomial ideal in a standard graded polynomial ring.
Example
R = QQ[v..z]
I = ideal(v*w*x,x*y,y*z)
Text
This is the Stanley-Reisner ideal of a simplicial complex $\Delta$
whose $f$-vector we compute below.
Example
needsPackage "SimplicialComplexes"
Δ = simplicialComplex I
f = matrix{fVector(Δ)}
Text
Next, we construct the ideal $\mathcal{P}_1 (I)$ of principal 1-jets of $I$
(see @TO "principalComponent"@ for details).
This is also the Stanley-Reisner ideal of a simplicial complex
$\Gamma_1$ and we can compute its $f$-vector.
Example
P1 = principalComponent(1,I)
phi = last flattenRing ring P1;
Γ1 = simplicialComplex phi P1
F = matrix{fVector Γ1}
Text
The $f$-vector of $\Gamma_1$ can be obtained by multiplying
the $f$-vector of $\Delta$ with a @TO "liftingMatrix"@ of the
appropriate size.
Example
L = liftingMatrix(1,4,7)
F == f*L
Text
There is a similar relation between the Betti numbers of
the Stanley-Reisner rings $\Bbbk [\Delta]$
and $\Bbbk [\Gamma_1]$. First, we compute the Betti
diagram of $\Bbbk [\Delta]$ and turn it into a matrix by
sliding the $i$-th row $i$ units to the right.
Example
betti res I
b = mutableMatrix(ZZ,3,5);
scanPairs(betti res I, (k,v) -> b_(k_2-k_0,k_2) = v);
b = matrix b
Text
Next, we do the same with the Betti diagram of $\Bbbk [\Gamma_1]$.
Example
betti res P1
B = mutableMatrix(ZZ,3,9);
scanPairs(betti res P1, (k,v) -> B_(k_2-k_0,k_2) = v);
B = matrix B
Text
The matrix containing the Betti numbers of $\Bbbk [\Gamma_1]$
can be obtained by multiplying the matrix containing the Betti
numbers of $\Bbbk [\Delta]$ with a @TO "liftingMatrix"@ of the
appropriate size.
Example
L = liftingMatrix(1,5,9)
B == b*L
///
end
5 changes: 5 additions & 0 deletions M2/Macaulay2/packages/Macaulay2Doc/changes.m2
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Expand Up @@ -47,6 +47,11 @@ document {
UL {
LI { "The function ", TO remove, ", which previously had no return value, now returns the value that was removed." }
}
},
LI { "improved packages:",
UL {
LI { TO "Jets::Jets", " has been updated to version 1.2 with improvements and new methods for principal jets."},
}
}
}
}
Expand Down

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