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Fix some issues raised in code review + epsilon #403

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b5676f5
kernel and wedge product docs fixes
joel-dodge May 22, 2026
0cb57c6
revert PushForward changes to docs
joel-dodge May 22, 2026
15a2611
fix a few random docs errors to get build working
joel-dodge May 22, 2026
615cf7a
use spaces not tabs
joel-dodge May 28, 2026
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71 changes: 40 additions & 31 deletions M2/Macaulay2/packages/Macaulay2Doc/functions/kernel-doc.m2
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Expand Up @@ -15,38 +15,47 @@ Node
syz
///

document {
Key => {(kernel,Matrix),
(kernel, RingElement)},
Headline => "kernel of a map of modules",
Usage => "kernel f, kernel a",
Inputs => {
"f" => {"a map of modules ", TT "M --> N"}
},
Outputs => {
Module => {"the kernel of f, a submodule of M"}
},
PARA{},
"The kernel is the submodule of M of all elements mapping to zero under ", TT "f", ".",
"If f is a RingElement it is interpreted as a 1 by 1 matrix",".",
EXAMPLE lines ///
R = ZZ/32003[a,b]/(ideal(a,b))^3
M = R^1/(ideal a^2)
mat = matrix{{a^2,b^2},{b,a}}
ker mat
presentation ker mat
syz mat
f = map(M++M, M++M, mat)
ker f
///,
SeeAlso => {
syz,
cokernel,
image,
map,
doc ///
Key
(kernel,Matrix)
(kernel,RingElement)
Headline
kernel of a map of modules
Usage
kernel f
kernel a
Inputs
f:Matrix
a map of modules $f:M \rightarrow N$.
Outputs
:Module
the kernel of $f$, a submodule of $M$
Description
Text
The kernel of $f$ is the submodule of $M$ consisting of all elements that go to zero under $f$.
Example
R = ZZ/32003[a,b]/(ideal(a,b))^3
M = R^1/(ideal a^2)
mat = matrix{{a^2,b^2},{b,a}}
ker mat
presentation ker mat
syz mat
f = map(M++M, M++M, mat)
ker f
Text
If $a$ is a @TO RingElement@, then $\text{kernel } a$ is the
kernel of the $1\times 1$ matrix giving multiplication by $a$:
$R^1 \rightarrow R^1$.
Example
kernel(a + b)
kernel((a + b) * id_(R^1))
SeeAlso
syz
cokernel
image
map
matrix
}
}
///

doc ///
Key
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2 changes: 1 addition & 1 deletion M2/Macaulay2/packages/Macaulay2Doc/functions/wedgeProduct-doc.m2
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@@ -1,7 +1,7 @@
doc ///
Key
(wedgeProduct, ZZ, ZZ, Module)
wedgeProduct
(wedgeProduct, ZZ, ZZ, Module)
Headline
the exterior multiplication map
Usage
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106 changes: 53 additions & 53 deletions M2/Macaulay2/packages/NumericalAlgebraicGeometry/doc.m2
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Expand Up @@ -930,41 +930,41 @@ document {
*-

doc ///
Key
evaluateH
evaluateHx
evaluateHt
Key
evaluateH
evaluateHx
evaluateHt

(evaluateH,Homotopy,Matrix,Number)
(evaluateHx,Homotopy,Matrix,Number)
(evaluateHt,Homotopy,Matrix,Number)
(evaluateH,Homotopy,Matrix,Number)
(evaluateHx,Homotopy,Matrix,Number)
(evaluateHt,Homotopy,Matrix,Number)

(evaluateH,ParameterHomotopy,Matrix,Matrix,Number)
(evaluateHx,ParameterHomotopy,Matrix,Matrix,Number)
(evaluateHt,ParameterHomotopy,Matrix,Matrix,Number)
(evaluateH,ParameterHomotopy,Matrix,Matrix,Number)
(evaluateHx,ParameterHomotopy,Matrix,Matrix,Number)
(evaluateHt,ParameterHomotopy,Matrix,Matrix,Number)

(evaluateH,SpecializedParameterHomotopy,Matrix,Number)
(evaluateHx,SpecializedParameterHomotopy,Matrix,Number)
(evaluateHt,SpecializedParameterHomotopy,Matrix,Number)
(evaluateH,SpecializedParameterHomotopy,Matrix,Number)
(evaluateHx,SpecializedParameterHomotopy,Matrix,Number)
(evaluateHt,SpecializedParameterHomotopy,Matrix,Number)

(evaluateH,GateHomotopy,Matrix,Number)
(evaluateHx,GateHomotopy,Matrix,Number)
(evaluateHt,GateHomotopy,Matrix,Number)
(evaluateH,GateHomotopy,Matrix,Number)
(evaluateHx,GateHomotopy,Matrix,Number)
(evaluateHt,GateHomotopy,Matrix,Number)

Headline
evaluate a homotopy and its derivatives
Headline
evaluate a homotopy and its derivatives

Usage
evaluateH(H,x,t)
evaluateHx(H,x,t)
evaluateHt(H,x,t)
Usage
evaluateH(H,x,t)
evaluateHx(H,x,t)
evaluateHt(H,x,t)

evaluateH(H,p,x,t)
evaluateHx(H,p,x,t)
evaluateHt(H,p,x,t)
evaluateH(H,p,x,t)
evaluateHx(H,p,x,t)
evaluateHt(H,p,x,t)

Description
Text
Description
Text
These methods evaluate a homotopy and its derivatives at a given point.

@TO evaluateH@ evaluates the homotopy itself,
Expand All @@ -975,54 +975,54 @@ doc ///
For a @TO ParameterHomotopy@, parameter values must also be supplied.

Example
variables = declareVariable \ {x,y}
T = inputGate symbol T
variables = declareVariable \ {x,y}
T = inputGate symbol T

F = matrix{{x^2+y^2-1},{x*y}}
G = matrix{{x^2-y-1},{x+y-1}}
F = matrix{{x^2+y^2-1},{x*y}}
G = matrix{{x^2-y-1},{x+y-1}}

H = gateHomotopy((1-T)*F + T*G, matrix{{x,y}}, T)
H = gateHomotopy((1-T)*F + T*G, matrix{{x,y}}, T)

X0 = matrix{{.2},{0}}
X0 = matrix{{.2},{0}}

evaluateH(H,X0,0.5)
evaluateHx(H,X0,0.5)
evaluateHt(H,X0,0.5)
evaluateH(H,X0,0.5)
evaluateHx(H,X0,0.5)
evaluateHt(H,X0,0.5)
Text
For a @TO ParameterHomotopy@, parameter values must also be supplied.

Example
variables = declareVariable \ {x,y}
params = declareVariable \ {a,b}
variables = declareVariable \ {x,y}
params = declareVariable \ {a,b}

F = gateSystem(
F = gateSystem(
matrix{params},
matrix{variables},
matrix{{a*x*y-1},{x^3+y^2-b}}
)
)

PH = parametricSegmentHomotopy F
PH = parametricSegmentHomotopy F

pStart = matrix{{1,2}}
pTarget = matrix{{2,1}}
pStart = matrix{{1,2}}
pTarget = matrix{{2,1}}

pars = transpose(pStart | pTarget)
pars = transpose(pStart | pTarget)

X0 = matrix{{.5},{1}}
X0 = matrix{{.5},{1}}

evaluateH(PH,pars,X0,0.5)
evaluateHx(PH,pars,X0,0.5)
evaluateHt(PH,pars,X0,0.5)
evaluateH(PH,pars,X0,0.5)
evaluateHx(PH,pars,X0,0.5)
evaluateHt(PH,pars,X0,0.5)
Text
After the parameter homotopy is specialized using @TO specialize@,
For a @TO ParameterHomotopy@, parameter values must also be supplied.
For a @TO ParameterHomotopy@, parameter values must also be supplied.

Example
H01 = specialize(PH, pars)
H01 = specialize(PH, pars)

evaluateH(H01,X0,0.5)
evaluateHx(H01,X0,0.5)
evaluateHt(H01,X0,0.5)
evaluateH(H01,X0,0.5)
evaluateHx(H01,X0,0.5)
evaluateHt(H01,X0,0.5)
///

document {
Expand Down
28 changes: 1 addition & 27 deletions M2/Macaulay2/packages/PushForward.m2
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Expand Up @@ -296,23 +296,10 @@ doc ///

This package was originally implemented by Claudiu Raicu, some changes were
introduced by Karl Schwede, and later by David Eisenbud and Mike Stillman.
Text
Start with a finite ring map and a module over the target ring.
Example
kk = QQ;
T = kk[t];
S = kk[a,b]/ideal(a*b);
use S;
M = module ideal(a);
phi = map(S,T,{a});
pushFwd(phi, M)
Subnodes
(pushFwd, RingMap)
(pushFwd, RingMap, Module)
(pushFwd, RingMap, Matrix)
SeeAlso
pushFwd
isModuleFinite
///

doc ///
Expand Down Expand Up @@ -458,24 +445,11 @@ document{
Key => pushFwd,
Headline => "push forward",
"The push forward functor",
PARA {
"Start with a finite ring map and a module over the target ring."
},
EXAMPLE lines ///
kk = QQ;
T = kk[t];
S = kk[a,b]/ideal(a*b);
use S;
M = module ideal(a);
phi = map(S,T,{a});
pushFwd(phi, M)
///,
UL {
{TO (pushFwd,RingMap)," - for a finite ring map"},
{TO (pushFwd,RingMap,Module), " - for a module"},
{TO (pushFwd,RingMap,Matrix), " - for a map of modules"}
},
SeeAlso => {(pushFwd,RingMap),(pushFwd,RingMap,Module),(pushFwd,RingMap,Matrix),isModuleFinite}
}
}

-- document{
Expand Down
65 changes: 33 additions & 32 deletions M2/Macaulay2/packages/SymbolicPowers.m2
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Expand Up @@ -1527,20 +1527,21 @@ doc ///


doc ///
Key
SampleSize
Headline
optional parameter used for approximating asymptotic invariants that are defined as limits.
Example
R = QQ[x,y,z];
J = ideal (x*(y^3-z^3),y*(z^3-x^3),z*(x^3-y^3));
waldschmidt(J, SampleSize=>5)
asymptoticRegularity(J, SampleSize=>5)
SeeAlso
waldschmidt
lowerBoundResurgence
asymptoticRegularity
///
Key
SampleSize
Headline
optional parameter used for approximating asymptotic invariants that are defined as limits.
Description
Example
R = QQ[x,y,z];
J = ideal (x*(y^3-z^3),y*(z^3-x^3),z*(x^3-y^3));
waldschmidt(J, SampleSize=>5)
asymptoticRegularity(J, SampleSize=>5)
SeeAlso
waldschmidt
lowerBoundResurgence
asymptoticRegularity
///

doc ///
Key
Expand Down Expand Up @@ -1583,27 +1584,27 @@ doc ///


doc ///
Key
InSymbolic
Headline
an optional parameter used in containmentProblem.
Example
R = QQ[x,y,z];
J = ideal (x*(y^3-z^3),y*(z^3-x^3),z*(x^3-y^3));
containmentProblem(J,3,InSymbolic => true)
SeeAlso
containmentProblem

///
Key
InSymbolic
Headline
an optional parameter used in containmentProblem.
Description
Example
R = QQ[x,y,z];
J = ideal (x*(y^3-z^3),y*(z^3-x^3),z*(x^3-y^3));
containmentProblem(J,3,InSymbolic => true)
SeeAlso
containmentProblem
///

doc ///
Key
Key
[containmentProblem,InSymbolic]
Headline
Headline
an optional parameter used in containmentProblem.
Usage
Usage
containmentProblem(I,n,InSymbolic => true)
Description
Description
Text
Given an ideal I and an integer n, @TO InSymbolic@ is used to ask the following question:
What is the largest power containing the symbolic power $I^{(n)}$?
Expand All @@ -1612,8 +1613,8 @@ doc ///
R = QQ[x,y,z];
J = ideal (x*(y^3-z^3),y*(z^3-x^3),z*(x^3-y^3));
containmentProblem(J,3,InSymbolic => true)
///
///

doc ///
Key
asymptoticRegularity
Expand Down