--------------------------------------------------------------------------------
-- Copyright 2021 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
-- Foundation, either version 3 of the License, or (at your option) any later
-- version.
--
-- This program is distributed in the hope that it will be useful, but WITHOUT
-- ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
-- FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
-- details.
--
-- You should have received a copy of the GNU General Public License along with
-- this program. If not, see .
--------------------------------------------------------------------------------
newPackage(
"Jets",
Version => "1.1",
Date => "June 10, 2022",
AuxiliaryFiles => true,
Authors => {
{
Name => "Federico Galetto",
Email => "galetto.federico@gmail.com",
HomePage => "http://math.galetto.org"
},
{
Name=> "Nicholas Iammarino",
Email=> "nickiammarino@gmail.com"
}
},
Headline => "compute jets of various algebraic, geometric and combinatorial objects",
PackageImports => {"SimpleDoc","EdgeIdeals"},
PackageExports => {"EdgeIdeals"},
DebuggingMode => false
)
importFrom(MinimalPrimes, {"radical"});
export {
"JJ",
"jets",
"jetsMaxOrder",
"jetsBase",
"jetsRing",
"projet",
"jet",
"jetsMatrix",
"jetsRadical",
"jetsProjection",
"jetsInfo",
"principalComponent",
"Saturate"
}
jetsOptions = {
Projective=> false
-- these are set up in case one needs to pass these options
-- to jets of a RingMap
-- DegreeMap=> null,
-- DegreeLift=> null
};
---------------------------------------------------------------------------
--helpers------------------------------------------------------------------
---------------------------------------------------------------------------
--create new-tier variables for jets ring
--by appending the order n as a string to the variable names
-*
jetsVariables = (n,R) -> (
symList := apply(gens R, baseName);
nString := toString n;
varNames:=
for s in symList list (
if instance(s,IndexedVariable) then (
name := separate("_", toString s);
name#0 | nString | "_" | name#1
) else (
toString s | nString
)
);
varNames = apply(varNames,value)
)
*-
jetsVariables= (n,R) -> (
symList := apply(gens R, baseName);
nString := toString n;
for s in symList list (
if instance(s,IndexedVariable) then (
name := (toString s#0) | nString;
(getSymbol name)_(s#1)
) else (
getSymbol (toString s | nString)
)
)
)
--generate degree list for order n jets variables
--this is used to create the rings of projective jets
degGenerator = (n,R) -> apply(degrees R, d -> toList((#d):n))
--generate degrees/map for truncation ring in ideal calculation
jetsDegrees = jetsOptions >> o -> R -> (
Tdegrees := null;
degreeMap := null;
if o.Projective then (
Tdegrees = -1* {degree R_0};
degreeMap = d -> degree 1_R;
) else (
Tdegrees = {degree 1_R};
degreeMap = identity;
);
(Tdegrees, degreeMap)
)
--------------------------------------------------------------------------
--method functions--------------------------------------------------------
--------------------------------------------------------------------------
--Jets (Main Method)------------------------------------------------------
jets = method(Options=>jetsOptions);
jets(ZZ,PolynomialRing) := PolynomialRing => o -> (n,R) -> (
if n<0 then error("jets order must be a non-negative integer");
if not isCommutative R then error("jets method does not support noncommutative rings");
--name to assign "storage" hashtable to be cached in the base ring
typeName := if o.Projective then (projet) else (jet);
jetDegs := null;--initialize degree list for jets variables
if not R#? typeName then (
jetDegs = if o.Projective then degGenerator(0, R) else degrees R;
R#typeName = new CacheTable from {
(symbol jetsMaxOrder)=> 0,
(symbol jetsRing)=> coefficientRing R[jetsVariables(0,R),
Join=> false,
Degrees=> jetDegs],
}
);
m := R#typeName#jetsMaxOrder;
S := R#typeName#jetsRing;
--build jet ring tower incrementally up to order n
if n>m then (
for i from m+1 to n do(
jetDegs = if o.Projective then degGenerator(i,R) else degrees R;
S = S[jetsVariables(i,R),
Join=> false,
Degrees=> jetDegs];
);
R#typeName#jetsMaxOrder = n;
R#typeName#jetsRing = S;
) else if m>n then (
for i from 0 to m-n-1 do (
S = coefficientRing S;
)
);
S#jetsInfo = new CacheTable from {
(symbol jetsBase)=> R,
(symbol Projective)=> o.Projective
};
S
)
jets(ZZ,Ideal) := Ideal => o -> (n,I) -> (
if n<0 then error("jets order must be a non-negative integer");
R := ring I;
S := null;--initializes jets ring
t := local t;--initializes truncation variable
typeName := if o.Projective then (projet) else (jet);
if not I.cache#? typeName then (
S = jets(0,R, Projective=> o.Projective);
I.cache#typeName = new CacheTable from {
(symbol jetsMaxOrder)=> 0,
(symbol jetsMatrix)=> (map(S,R,vars S)) gens I
};
);
m := I.cache#typeName#jetsMaxOrder;
--calculate higher order entries if needed
if n>m then (
S = jets(n,R, Projective=> o.Projective);
(Tdegrees, degreeMap) := jetsDegrees (R, Projective=> o.Projective);
T := S[t, Degrees=> Tdegrees, Join=> false]/(ideal(t^(n+1)));
--a row matrix of substitution polynomials in t with coefficients
--in the jets ring. Calculated incrementally from variables of each
--level of the tower.
tempS := S;
Tpolys := sum join(
(for i from 0 to n-1 list(
promote(matrix t^(n-i),T) * vars tempS
) do (
tempS = coefficientRing tempS)),
{promote (matrix t^0,T) * vars tempS}
);
phi := map(T,R,Tpolys,DegreeMap=> degreeMap);
--a list of generators for I is obtained to avoid dropping/repeating
geners := I_*;
--condition determining if all generators of the ideal are constants
constCond := all(geners,isConstant);
--add dummy generator to avoid loss of zeros
gensI := if constCond then matrix{geners | {R_0}} else matrix{geners};
c := last coefficients(phi gensI);
--remove dummy generators if necessary
if constCond then c = c_{0..(numColumns c - 2)};
resultMatrix := lift(c,S);
--update value in ideal cache
I.cache#typeName#jetsMatrix = resultMatrix;
I.cache#typeName#jetsMaxOrder = n;
m=n;
);
--retrieve ideal of appropriate order
JMatrix := I.cache#typeName#jetsMatrix;
if zero JMatrix then return ideal(0_(jets(n,R)));
f := map(jets(n,R,Projective=> o.Projective),jets(m,R, Projective=> o.Projective));
J := f ideal (JMatrix^{m-n..m});
J.cache#jetsInfo = new CacheTable from {
jetsBase=> I,
Projective=> o.Projective
};
J
)
jets(ZZ,QuotientRing) := QuotientRing => o -> (n,R) -> (
if n<0 then error("jets order must be a non-negative integer");
splitQuotient := presentation R;
ambientRing := ring splitQuotient;
base := null; --jets ring to be used in quotient
modI := null; --jets ideal to be used in quotient
Q := null; --variable to store quotient ring
typeName := if o.Projective then (projet) else (jet);
if not R#? typeName then (
base = jets(0, ambientRing, Projective=> o.Projective);
modI = jets(0, ideal(splitQuotient), Projective=> o.Projective);
R#typeName = new CacheTable from {
(symbol jetsRing)=> new CacheTable from {
0 => base/modI
},
};
);
--form the jets of a quotient ring by taking the quotients of a jets
--ring and a jets ideal. Each order of the quotient is stored in a
--cache table with the integer value of the order as the key
if R#typeName#jetsRing#? n then (
Q = R#typeName#jetsRing#n;
) else (
base = jets(n, ambientRing, Projective=> o.Projective);
modI = jets(n, ideal(splitQuotient), Projective=> o.Projective);
Q = base/modI;
R#typeName#jetsRing#n = Q;
Q#jetsInfo = new CacheTable from {
jetsBase=> R,
Projective=> o.Projective
}
);
Q
)
jets(ZZ,RingMap) := RingMap => o -> (n,phi) -> (
if n<0 then error("jets order must be a non-negative integer");
I := ideal(phi.matrix);
typeName := if o.Projective then (projet) else (jet);
-- check whether jets have been calculated for this map
if (not phi.cache#? typeName) then (
jets(0,I, Projective=> o.Projective);
phi.cache#typeName = new CacheTable from {
(symbol jetsMaxOrder)=> 0,
(symbol jetsMatrix)=> (map(jets(0,phi.target, Projective=> o.Projective),
jets(0,phi.source, Projective=> o.Projective),
I.cache#typeName#jetsMatrix)).matrix
};
);
JR := jets(n,phi.source, Projective=> o.Projective);
JS := jets(n,phi.target, Projective=> o.Projective);
targets := null;
--check whether lower order jets have already been calculated
m := phi.cache#typeName#jetsMaxOrder;
if m < n then (
jets(n,I, Projective=> o.Projective);
targets = (I.cache#typeName#jetsMatrix);
phi.cache#typeName#jetsMaxOrder = n;
phi.cache#typeName#jetsMatrix = targets;
) else (
targets = phi.cache#typeName#jetsMatrix^{m-n..m};
--need to lift 'targets' to jets of order m-n
targets=lift(targets,JS);
);
psi := map(JS,JR,flatten transpose targets);
psi.cache#jetsInfo = new CacheTable from {
jetsBase=> phi,
Projective=> o.Projective
};
psi
)
jets(ZZ,Graph) := Graph => o -> (n,G) -> (
if n<0 then error("jets order must be a non-negative integer");
--get the list of edges of the jets of the (hyper)graph
--ring is flattened because graphs don't play well with towers of rings
E := (flattenRing(jetsRadical(n,edgeIdeal G),Result=>1)) / support;
--create graph
graph E
)
jets(ZZ,HyperGraph) := HyperGraph => o -> (n,G) -> (
if n<0 then error("jets order must be a non-negative integer");
--get the list of edges of the jets of the (hyper)graph
--ring is flattened because graphs don't play well with towers of rings
E := (flattenRing(jetsRadical(n,edgeIdeal G),Result=>1)) / support;
--create hypergraph
hyperGraph E
)
jets(ZZ,AffineVariety) := 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);
if o.Projective then return Proj JR else return Spec JR;
)
---Secondary Methods--------------------------------------------------
--to potentially reduce computation time for monomial jet ideals
--(see documentation)
jetsRadical = method(TypicalValue=>Ideal);
jetsRadical(ZZ,Ideal) := (n,I) -> (
if n<0 then error("jets order must be a non-negative integer");
if isMonomialIdeal I then (
baseIdeal := jets(n,I);
R := ring I;
gensList := flatten entries gens baseIdeal;
termList := apply(gensList, t-> terms(coefficientRing R, t));
squarefreeGens := apply(apply(flatten termList, support),product);
ideal(squarefreeGens)
) else (
radical jets(n,I)
)
)
--to create a map sending elements of a jets ring to a jets ring of
--higher order
jetsProjection = method(Options=>jetsOptions,TypicalValue=>RingMap);
jetsProjection(ZZ,ZZ,PolynomialRing) :=
jetsProjection(ZZ,ZZ,QuotientRing) := o -> (t,s,R) -> (
if t < s then error("first argument must be less than or equal to the second");
if t<0 or s<0 then error("jets orders must be non-negative integers");
(map(jets(t,R,Projective=> o.Projective),jets(s,R,Projective=> o.Projective)))
)
--scripted functor for jets
--this modeled after the code for Tor
--if new jets methods are added, this will automatically work
JJ = new ScriptedFunctor from {
subscript => (
i -> new ScriptedFunctor from {
argument => (X -> (
jetsOptions >> o -> Y -> (
f := lookup(jets,class i,class Y);
if f === null then error "no method available"
else (f o)(i,Y)
)
) X
)
}
)
}
--compute an ideal whose vanishing locus is the
--principal component of the jets of an ideal
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");
-- 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
--then to the n jets via the map p
sing := p(i(ideal singularLocus I));
--default is to saturate JI wrt sing
if o.Saturate then (
saturate(JI,sing)
)
--if JI is radical, colon is enough
else (
JI:sing
)
)
beginDocumentation()
----------------------------------------------------------------------
-- TESTS
----------------------------------------------------------------------
TEST ///
R = QQ[x,y,z];
assert(degrees jets(2,R) === {{1}, {1}, {1}})
assert(degrees jets(2,R,Projective=> true) === {{2}, {2}, {2}})
I=ideal(y-x^2,z-x^3);
assert(not(isHomogeneous jets(2,I)))
assert(isHomogeneous jets(2,I,Projective=>true))
///
--for non uniform degrees
TEST ///
R = QQ[x,y,z, Degrees=> {2,3,1}];
assert(degrees jets(2,R) === {{2}, {3}, {1}})
assert(degrees jets(2,R,Projective=> true) === {{2}, {2}, {2}})
I = ideal(x*y, x*z^2);
J = ideal(x^3-y*z^3, y+x*z);
assert(isHomogeneous jets(2,I))
assert(isHomogeneous jets(2,I,Projective=>true))
assert(isHomogeneous jets(2,J))
assert(isHomogeneous jets(2,J,Projective=>true))
X = radical jets(2,I);
Y = jetsRadical(2,I);
assert(X == Y)
assert(mingens X === mingens Y);
///
TEST ///
R=QQ[x,y, Degrees=> {2,3}];
S=QQ[a,b,c, Degrees=> {1,1,2}]
phi = map(S,R, {a^2 + c, b*c});
f = jets(2,phi);
testx = c2+2*a0*a2+a1^2;
testy = b0*c2+c0*b2+b1*c1;
assert(f x2 === testx)
assert(f y2 === testy)
assert(isHomogeneous jets(3,phi))
assert(isHomogeneous jets(3,phi,Projective=>true))
///
--for ideals with constant generators
TEST ///
R=QQ[x]
I0 = ideal(2_R)
Ftest0=jets(2,I0)
assert(Ftest0 == jets(2,R))
I1 = ideal(2_R,x)
Ftest1=jets(2,I1)
assert(Ftest1 == jets(2,R))
S=ZZ[x]
J0 = ideal(2_S)
Ztest0 = jets(2,J0)
assert(Ztest0!=jets(2,S))
J1 = ideal(2_S,x)
Ztest1=jets(2,J1)
assert(Ztest1!=jets(2,S))
///
--for principal component
TEST ///
R=QQ[x,y]
I=ideal(y^2-x^3)
PC=principalComponent(2,I)
P=primaryDecomposition jets(2,I)
C=first select(P,c -> degree c == 6)
assert(PC == C)
///
--for quotients and varieties
TEST ///
R = QQ[x,y]
I = ideal(y^2,x^3)
Q = R/I
JR = jets(2,R)
JI = jets(2,I)
JQ = jets(2,Q)
assert(JR === ambient JQ)
assert(JI === ideal JQ)
assert(presentation (JR/JI) === presentation JQ)
V = Spec Q
JV = jets(2,V)
assert(ring JV === JQ)
///
--for graphs
TEST ///
R=QQ[x,y,z]
G = graph(R,{{x,y},{y,z},{x,z}})
JG = jets(1,G)
JR = jets(1,R)
use ring JG
test = {{x0,y0},{x0,z0},{y0,z0},{x1,y0},{x1,z0},{y1,x0},{y1,z0},{z1,x0},{z1,y0}}
assert((set edges JG) === (set test))
///
--for projections
TEST ///
R=QQ[x,y,z]
I = ideal(y-x^2,z-x^3)
JI = jets(1,I)
p = jetsProjection(3,1,R)
assert(ring p JI === jets(3,R))
///
----------------------------------------------------------------------
-- Documentation
----------------------------------------------------------------------
doc ///
Node
Key
Jets
Headline
compute jets of various algebraic, geometric and combinatorial objects
Description
Text
This package enables computations with jet functors.
It introduces the @TO jets@ method to compute jets of
polynomial rings, ideals, quotients, ring homomorphisms,
and varieties.
The construction of jets follows an algebraic procedure
discussed in several sources, including the first three
references below.
Additional features include an alternative algorithm to compute
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.
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.")@
@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.")@
Subnodes
:Package methods
jets
jetsProjection
jetsRadical
principalComponent
:Examples from the literature
"Example 1"
"Example 2"
"Example 3"
:Technical information
"Storing Computations"
Node
Key
jets
Headline
compute the jets of an object
Subnodes
(jets,ZZ,PolynomialRing)
(jets,ZZ,Ideal)
(jets,ZZ,QuotientRing)
(jets,ZZ,RingMap)
(jets,ZZ,Graph)
(jets,ZZ,AffineVariety)
[jets,Projective]
JJ
Node
Key
"Storing Computations"
Headline
caching scheme for jets computations
Description
Text
In many cases, the @TO jets@ method will store its results inside
a @TO CacheTable@ in the base object. When the method is called
again with the same or a lower jets order, the result is pulled
from the cache.
For polynomial rings, data is stored under @TT "*.jet"@.
Example
R = QQ[x,y]
R.?jet
jets(3,R)
R.?jet
peek R.jet
Text
Note also that rings of jets are built as towers from lower to
higher jets orders. Therefore it is possible to store a single
ring of the highest order computed thus far.
For ideals, data is stored under @TT "*.cache.jet"@.
A single matrix is stored containing generators for the
highest order of jets computed thus far.
Generators for lower orders are recovered from this matrix
without additional computations.
Example
I = ideal (x^2 - y)
I.cache.?jet
elapsedTime jets(3,I)
I.cache.?jet
peek I.cache.jet
elapsedTime jets(3,I)
elapsedTime jets(2,I)
Text
For quotient rings, data is stored under @TT "*.jet"@.
Each jets order gives rise to a different quotient
that is stored separately under @TT "*.jet.jetsRing"@
(order zero jets are always included by default).
Example
Q = R/I
Q.?jet
jets(3,Q)
Q.?jet
peek Q.jet.jetsRing
jets(2,Q)
peek Q.jet.jetsRing
Text
For ring homomorphisms, data is stored under @TT "*.cache.jet"@.
A single matrix is stored describing the map for the
highest order of jets computed thus far.
Lower orders map are recovered from this matrix
without additional computations.
Example
S = QQ[t]
f = map(S,Q,{t,t^2})
isWellDefined f
f.cache.?jet
elapsedTime jets(3,f)
f.cache.?jet
peek f.cache.jet
elapsedTime jets(2,f)
Text
Projective jets data is stored separately under @TT "*.projet"@
or @TT "*.cache.projet"@ to accommodate for the different grading.
Example
jets(2,I,Projective=>true)
peek I.cache.projet
peek R.projet
Caveat
No data is cached when computing jets of affine varieties and (hyper)graphs,
radicals, or principal components.
Subnodes
jet
projet
jetsRing
jetsMaxOrder
jetsMatrix
jetsBase
jetsInfo
Node
Key
(jets,ZZ,PolynomialRing)
Headline
compute jets of a polynomial ring
Usage
jets (n,R)
Inputs
n:ZZ
R:PolynomialRing
Outputs
:PolynomialRing
of jets order @TT "n"@.
Description
Text
This function is provided by the package @TO Jets@. Rings are
constructed incrementally as towers. The function returns the
ring with variables in the jets order requested, and coeffients
in all lower orders. The grading or multigrading of the jets ring
follows from that of the base ring.
Example
R = QQ[x,y,z,Degrees=>{2,1,3}]
JR = jets(2,R)
describe JR
degrees (flattenRing JR)_0
Text
When the @TO [jets,Projective]@ option is set to true, the degree
of each jets variable matches the jets order, in accordance with
Proposition 6.6 (c) of @arXiv("math/0407113","P. Vojta,
Jets via Hasse-Schmidt Derivations")@.
Example
R = QQ[x,y,z,Degrees=>{2,1,3}]
JR = jets(2,R,Projective=>true)
degrees (flattenRing JR)_0
Text
The convention for labeling variables in the jets of polynomial ring
is to append the order of the jets to name of the variables in the
base ring. Existing subscripts are preserved.
Example
A = QQ[a_1..a_3]
JA = jets(1,A)
describe JA
Text
Note that the coefficient ring of the polynomial ring does not need
to be a field. The jets of the input polynomial ring will be a
polynomial ring with the same coefficient ring as the input.
Example
Zi = ZZ[i]/ideal(i^2+1)
B = Zi[b_1..b_3]
JB = jets(1,B)
describe JB
Caveat
With @TT "Projective=>true"@ the jet variables of order zero have degree 0,
therefore no heft vector exist for the ambient ring of the jets.
As a result, certain computations will not be supported, and others may run more slowly.
See @TO "Macaulay2Doc::heft vectors"@ for more information.
Node
Key
(jets,ZZ,Ideal)
Headline
compute jets of a an ideal in a polynomial ring
Usage
jets (n,I)
Inputs
n:ZZ
I:Ideal
Outputs
:Ideal
generated by the jets of the generators of @TT "I"@
Description
Text
This function is provided by the package
@TO Jets@.
Example
R = QQ[x,y]
I = ideal (x^3 + y^3 - 3*x*y)
J = jets(3,I);
netList J_*
Text
When the @TO [jets,Projective]@ option is set to true, the degree
of each jets variable matches its order, in accordance with
Proposition 6.6 (c) of @arXiv("math/0407113","P. Vojta,
Jets via Hasse-Schmidt Derivations")@.
As a result, the jets of any ideal will be homogeneous regardless
of the homogeneity of the base ideal, or that of its affine jets.
Example
R = QQ[x,y,z]
I = ideal (y-x^2, z-x^3)
JI = jets(2,I)
isHomogeneous JI
JIproj = jets(2,I,Projective=>true)
isHomogeneous JIproj
Caveat
With @TT "Projective=>true"@ the jet variables of order zero have degree 0,
therefore no heft vector exist for the ambient ring of the jets.
As a result, certain computations will not be supported, and others may run more slowly.
See @TO "Macaulay2Doc::heft vectors"@ for more information.
Node
Key
(jets,ZZ,QuotientRing)
Headline
the jets of an affine algebra
Usage
jets (n,Q)
Inputs
n:ZZ
Q:QuotientRing
Outputs
:QuotientRing
Description
Text
This function is provided by the package @TO Jets@. Forms the jets of a @TO QuotientRing@ by forming the quotient of
@TO (jets,ZZ,PolynomialRing)@ of the ambient ring of @TT "Q"@ with
@TO (jets,ZZ,Ideal)@ of the ideal defining @TT "Q"@
Example
R = QQ[x,y];
I = ideal(y^2-x^3);
Q = R/I;
JQ = jets(2,Q);
describe JQ
Caveat
Forming quotients triggers a Groebner basis computation, which may be time consuming.
Node
Key
(jets,ZZ,RingMap)
Headline
the jets of a homomorphism of rings
Usage
jets (n,f)
Inputs
n:ZZ
f:RingMap
Outputs
:RingMap
obtained by applying the @TT "n"@-th jets functor to @TT "f"@
Description
Text
This function is provided by the package
@TO Jets@.
Example
R = QQ[x,y,z]
S = QQ[t]
f = map(S,R,{t,t^2,t^3})
Jf = jets(2,f);
matrix Jf
Text
This function can also be applied when the source and/or the target
of the ring homomorphism are quotients of a polynomial ring
Example
I = ideal(y-x^2,z-x^3)
Q = R/I
g = map(S,Q,{t,t^2,t^3})
isWellDefined g
Jg = jets(2,g);
isWellDefined Jg
Node
Key
(jets,ZZ,Graph)
(jets,ZZ,HyperGraph)
Headline
the jets of a graph
Usage
jets (n,G)
Inputs
n:ZZ
G:Graph
undirected, finite, and simple graph or hypergraph
Outputs
:Graph
the (hyper)graph of @TT "n"@-jets of @TT "G"@
Description
Text
This function is provided by the package
@TO Jets@.
Jets of graphs are defined in § 2 of
@arXiv("2104.08933","F. Galetto, E. Helmick, and M. Walsh,
Jet graphs")@.
The input is of type @TO "EdgeIdeals::Graph"@ as defined by
the @TO EdgeIdeals@ package, which is automatically exported
when loading @TO Jets@.
Example
R = QQ[x,y,z]
G = graph(R,{{x,y},{y,z}})
JG = jets(2,G)
vertexCovers JG
Text
We can also calculate the jets of a @TO "EdgeIdeals::HyperGraph"@.
Example
R = QQ[u,v,w,x,y,z]
H = hyperGraph(R,{{u},{v,w},{x,y,z}})
jets(1,H)
Caveat
Rings of jets are usually constructed as towers of rings with
tiers corresponding to jets of different orders. However, the
tower is flattened out before constructing the edge ideal of
the jets of a (hyper)graph. This is done in order to ensure
compatibility with the @TO "EdgeIdeals::EdgeIdeals"@ package.
Node
Key
(jets,ZZ,AffineVariety)
Headline
the jets of an affine variety
Usage
jets (n,V)
Inputs
n:ZZ
V:AffineVariety
Outputs
:Variety
an @TO AffineVariety@ or a @TO ProjectiveVariety@
Description
Text
Returns the jets of an @TO AffineVariety@ as an @TO AffineVariety@.
Example
R = QQ[x,y]
I = ideal(y^2-x^2*(x+1))
A = Spec(R/I)
jets(2,A)
Text
If @TO [jets,Projective]@ is set to true, then jets are computed
with the grading introduced in Proposition 6.6 (c) of
@arXiv("math/0407113","P. Vojta, Jets via Hasse-Schmidt Derivations")@,
and the function returns a @TO ProjectiveVariety@.
Example
jets(2,A,Projective=>true)
Caveat
With @TT "Projective=>true"@ the jet variables of order zero have degree 0,
therefore no heft vector exist for the ambient ring of the jets.
As a result, certain computations will not be supported, and others may run more slowly.
See @TO "Macaulay2Doc::heft vectors"@ for more information.
Note: jets of projective varieties are currently not implemented.
Node
Key
jet
Headline
hashtable key
Description
Text
@TO CacheTable@ for storing information on jets constructed
from a base object @TT "x"@.
For @TO PolynomialRing@ and @TO QuotientRing@, stored as @TT "x.*"@.
For @TO RingMap@ and @TO Ideal@ stored as @TT "x.cache.*"@.
SeeAlso
projet
jetsRing
jetsMaxOrder
jetsMatrix
jetsBase
jetsInfo
Node
Key
projet
Headline
hashtable key
Description
Text
@TO CacheTable@ for storing information on the projective jets
of the base object @TT "x"@.
For @TO PolynomialRing@ and @TO QuotientRing@, stored as @TT "x.*"@.
For @TO RingMap@ and @TO Ideal@ stored as @TT "x.cache.*"@.
SeeAlso
jet
jetsRing
jetsMaxOrder
jetsMatrix
jetsBase
jetsInfo
Node
Key
jetsRing
Headline
hashtable value
Description
Text
The @TO (jets,ZZ,PolynomialRing)@ of highest order computed thus far
for a particular base ring. Stored in @TO jet@ or @TO projet@
of the base.
SeeAlso
jet
projet
jetsMaxOrder
jetsMatrix
jetsBase
jetsInfo
Node
Key
jetsMatrix
Headline
hashtable value
Description
Text
A matrix of jets elements which generate a @TO (jets,ZZ,Ideal)@
or serve as targets for a @TO (jets,ZZ,RingMap)@. Stored in
@TO jet@ or @TO projet@ of the base.
SeeAlso
jet
projet
jetsRing
jetsMaxOrder
jetsBase
jetsInfo
Node
Key
jetsMaxOrder
Headline
hashtable value
Description
Text
The highest order of jets computed thus far for a particular
object. Stored in @TO jet@ or @TO projet@ of the base object.
SeeAlso
jet
projet
jetsRing
jetsMatrix
jetsBase
jetsInfo
Node
Key
jetsBase
Headline
hashtable value
Description
Text
The base ring of a @TO (jets,ZZ,PolynomialRing)@. Stored in a jets object
for reference.
SeeAlso
jet
projet
jetsRing
jetsMaxOrder
jetsMatrix
jetsInfo
Node
Key
jetsInfo
Headline
hashtable key
Description
Text
@TO CacheTable@ for storing information on the base object within
the jets object.
SeeAlso
jet
projet
jetsRing
jetsMatrix
jetsBase
jetsMaxOrder
Node
Key
[jets,Projective]
[jetsProjection,Projective]
Headline
Option for jets
Description
Text
Set the degree of each jet variable to match its order,
as in Proposition 6.6 (c) of
@arXiv("math/0407113","P. Vojta, Jets via Hasse-Schmidt Derivations")@.
This guarantees that the output of @TO jets@ is homogeneous.
Caveat
With @TT "Projective=>true"@ the jet variables of order zero have degree 0,
therefore no heft vector exist for the ambient ring of the jets.
As a result, certain computations will not be supported, and others may run more slowly.
See @TO "Macaulay2Doc::heft vectors"@ for more information.
SeeAlso
(jets,ZZ,PolynomialRing)
(jets,ZZ,Ideal)
Node
Key
jetsRadical
(jetsRadical,ZZ,Ideal)
Headline
compute radicals of jets ideals
Usage
jetsRadical(n,I)
Inputs
n:ZZ
I:Ideal
Outputs
:Ideal
radical of the nth jets of @TT "I"@
Description
Text
This function is provided by the package @TO Jets@. It returns the radical
of the ideal of jets of the input ideal.
If the input is not a monomial ideal, this function uses the @TO radical@ function from
the @TO MinimalPrimes@ package.
If the input is a monomial ideal, it uses an algorithm
based on the proof of Theorem 3.2 in @HREF("https://doi.org/10.1080/00927870500454927",
"R.A. Goward and K.E. Smith, The jet scheme of a monomial scheme")@.
This has the potential to speed up the computation, especially for large jet orders.
Note that the generating set of the output may not be minimal,
unless the generators of the input are squarefree monomials.
An ideal generated by squarefree monomials:
Example
R = QQ[x,y,z]
I = ideal (x*z, y*z)
J = jets(1,I);
MP = radical J;
GS = jetsRadical(1,I);
netList sort MP_* | netList sort GS_*
Text
An ideal with genereators which are not squarefree:
Example
R = QQ[x,y,z]
I = ideal(x*y^2, z*x, x^3)
J = jets(1,I);
MP = radical J;
GS = jetsRadical(1,I);
netList sort MP_* | netList sort GS_*
MP == GS
Node
Key
jetsProjection
(jetsProjection,ZZ,ZZ,PolynomialRing)
(jetsProjection,ZZ,ZZ,QuotientRing)
Headline
canonical map between jets rings
Usage
jets(t,s,R)
jets(t,s,Q)
Inputs
t:ZZ
s:ZZ
R:PolynomialRing
or a quotient of
Outputs
:RingMap
between jets orders
Description
Text
This function is provided by the package @TO Jets@. Generates an
inclusion map from the order @TT "s"@ into the order @TT "t"@ jets
of a (quotient of a) polynomial ring.
Throws an error if @TT "t~~false"@
to return the ideal $J\colon I$, which can speed up computations.
As an example, consider the union of three non parallel lines
in the affine plane. We compute the principal component of the
jets of order two.
Example
R = QQ[x,y]
I = ideal(x*y*(x+y-1))
PC = principalComponent(2,I)
Text
Despite the name, the principal component need not be a component
of the jet scheme (i.e., it need not be irreducible). In this example,
the principal component has degree 3 and is the union of three components
of degree 1.
Example
P = primaryDecomposition jets(2,I)
any(P,c -> c == PC)
PC == intersect(select(P,c -> degree c == 1))
Caveat
This function requires computation of a singular locus,
a saturation (or quotient), and jets, with each step being
potentially quite time consuming.
Subnodes
Saturate
Node
Key
Saturate
[principalComponent,Saturate]
Headline
option for principal components
Description
Text
Strategy for computing principal components of jet schemes
SeeAlso
principalComponent
Node
Key
JJ
Headline
scripted functor associated with jets
Usage
JJ_n X
Inputs
n:ZZ
Description
Text
Shorthand for @TO jets@
Example
R = QQ[x,y]
I = ideal(y^2-x^3)
JJ_2 R
JJ_2 I
Node
Key
"Example 1"
Headline
jets of monomial ideals
Description
Text
As observed in @HREF("https://doi.org/10.1080/00927870500454927",
"R.A. Goward and K.E. Smith, The jet scheme of a monomial scheme")@ [GS06],
the ideal of jets of a monomial ideal is typically not a monomial ideal.
Example
R = QQ[x,y,z]
I = ideal(x*y*z)
J2I = jets(2,I)
Text
However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial ideal. In
fact, the proof of [GS06, Theorem 3.2] shows that the radical is generated by the individual
terms in the generators of the ideal of jets. This observation provides an alternative
algorithm for computing radicals of jets of monomial ideals, which can be faster than the
default radical computation in Macaulay2.
Example
elapsedTime jetsRadical(2,I)
elapsedTime radical J2I
Text
For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes of the
ideal of jets. Moreover, the main theorem in @arXiv("math/0607638",
"C. Yuen, Multiplicity of jet schemes of monomial schemes")@ counts the multiplicity of the jet scheme
of a monomial hypersurface along its minimal primes (see also @HREF("https://doi.org/10.1080/00927870701512168",
"C. Yuen, The multiplicity of jet schemes of a simple normal crossing divisor")@). We compute the
minimal primes, then use the @TO "LocalRings::LocalRings"@ package to compute their multiplicities in
the second jet scheme of the example above. Note that we need to flatten the polynomial ring
of jets because the @TT "LocalRings"@ package does not allow towers of rings.
Example
P = minimalPrimes J2I
(A,f) = flattenRing ring J2I
needsPackage "LocalRings"
M = cokernel gens f J2I
mult = for p in P list (
Rp := localRing(A,f p);
length(M ** Rp)
);
netList(pack(4,mingle{P,mult}),HorizontalSpace=>1)
Node
Key
"Example 2"
Headline
jets of graphs
Description
Text
Jets of graphs were introduced in @arXiv("2104.08933","F. Galetto, E. Helmick, and M. Walsh,
Jet graphs")@ [GHW21]. Starting with a
finite, simple graph $G$, one may construct a quadratic squarefree
monomial ideal $I(G)$ (known as the \emph{edge ideal} of the graph) by
converting edges to monomials.
One may then consider the radical of the ideal of $s$-jets of $I(G)$,
which is again a quadratic squarefree monomial ideal. The graph
corresponding to this ideal is the graph of $s$-jets of $G$, denoted
$\mathcal{J}_s (G)$.
Jets of graphs and hypergraphs can be obtained by applying the
@TO jets@ method to objects of type @TO "EdgeIdeals::Graph"@ and
@TO "EdgeIdeals::HyperGraph"@ from the Macaulay2 @TO "EdgeIdeals::EdgeIdeals"@ package
(which is automatically loaded by the @TO Jets@ package.
Consider, for example, the graph in the figure below.
Code
IMG ("src" => replace("PKG", "Jets", currentLayout#"package") | "graph.png",
"alt" => "a graph on 5 vertices", "height" => "300")
Example
R = QQ[a..e]
G = graph({{a,c},{a,d},{a,e},{b,c},{b,d},{b,e},{c,e}})
Text
We compute the first and second order jets, and list their edges.
Example
J1G = jets(1,G); netList pack(7,edges J1G)
J2G = jets(2,G); netList pack(7,edges J2G)
Text
As predicted in [GHW21, Theorem 3.1], all jets have the same
chromatic number.
Example
apply({G,J1G,J2G},chromaticNumber)
Text
By contrast, jets may not preserve the property of being co-chordal.
Example
apply({G,J1G,J2G},x -> isChordal complementGraph x)
Text
Using Fröberg's Theorem (cf. R. Fröberg, On Stanley-Reisner rings),
we deduce that although
the edge ideal of a graph may have a linear free resolution, the edge
ideals of its jets may not have linear resolutions.
Finally, we compare minimal vertex covers of the graph and of its
second order jets.
Example
vertexCovers G
netList pack(2,vertexCovers J2G)
Text
With the exception of the second row, many vertex covers arise as
indicated in [GHW21, Proposition 5.2, 5.3].
Node
Key
"Example 3"
Headline
jets of determinantal varieties
Description
Text
Consider the determinantal varieties $X_r$ of
$3\times 3$ matrices of rank at most $r$, which are defined by the
vanishing of minors of size $r+1$. We illustrate computationally some
of the known results about jets.
Example
R = QQ[x_(1,1)..x_(3,3)]
G = genericMatrix(R,3,3)
Text
Since $X_0$ is a single point, its first jet scheme consists of a
single (smooth) point.
Example
I1 = minors(1,G)
JI1 = jets(1,I1)
dim JI1, isPrime JI1
Text
The jets of $X_2$ (the determinantal hypersurface) are known to be
irreducible (see Theorem 3.1 in @HREF("https://doi.org/10.1016/j.jpaa.2004.06.001",
"T. Košir, B.A. Sethuraman, Determinantal varieties over truncated polynomial rings")@ [KS05],
or Corollary 4.13 in @HREF("https://doi.org/10.1090/S0002-9947-2012-05564-4",
"R. Docampo, Arcs on determinantal varieties")@ [Doc13]).
Since $X_2$ is a complete intersection and has rational singularities
(see Corollary 6.1.5(b) in @HREF("https://doi.org/10.1017/CBO9780511546556",
"J. Weyman, Cohomology of vector bundles and syzygies")@),
this also follows from a more general result of M. Mustaţă
(Theorem 3.3 in @HREF("https://doi.org/10.1007/s002220100152",
"Jet schemes of locally complete intersection canonical singularities")@).
Example
I3 = minors(3,G)
JI3 = jets(1,I3)
isPrime JI3
Text
As for the case of $2\times 2$ minors, Theorem 5.1 in [KS05], Theorem 5.1 in
@arXiv("math/0608632","C. Yuen, Jet schemes of determinantal varieties")@,
and Corollary 4.13 in [Doc13] all count the number of components;
the first two references describe
the components further. As expected, the first jet scheme of $X_1$ has
two components, one of them an affine space.
Example
I2 = minors(2,G)
JI2 = jets(1,I2)
P = primaryDecomposition JI2; #P
P_1
Text
The other component is the so-called principal component of the jet
scheme, i.e., the Zariski closure of the first jets of the smooth
locus of $X_1$. To check this, we first establish that the first jet
scheme is reduced (i.e. its ideal is radical), then use the
@TO principalComponent@ method with the option
@TO [principalComponent,Saturate]@ set to @TT "false"@ to speed up computations.
Example
radical JI2 == JI2
P_0 == principalComponent(1,I2,Saturate=>false)
P_0
Text
Finally, as observed in Theorem 18 of @HREF("http://dx.doi.org/10.2140/pjm.2014.272.147",
"S.R. Ghorpade, B. Jonov and B.A. Sethuraman,
Hilbert series of certain jet schemes of determinantal varieties")@ the Hilbert
series of the principal component of the first jet scheme of $X_1$ is
the square of the Hilbert series of $X_1$.
Example
apply({P_0,I2}, X -> hilbertSeries(X,Reduce=>true))
numerator (first oo) == (numerator last oo)^2
///
end
~~