MultigradedImplicitization.m2

newPackage(
  "MultigradedImplicitization",
  Version => "1.0",
  Date => "October 26, 2023",
  Authors => {
    {Name => "Joseph Cummings",
    Email => "josephcummings03@gmail.com",
    HomePage => "https://sites.google.com/view/josephcummingsuky/home"},
    {Name => "Benjamin Hollering",
    Email => "benhollering@gmail.com",
    HomePage => "https://sites.google.com/view/benhollering"}
  },
  Headline => "A package for levaraging multigradings to solve implicitization problems",
  DebuggingMode => true,
  PackageImports => {"gfanInterface"}
)

--------------------
--Exports
--------------------

export {
  -- Methods
  "maxGrading",
  "trimBasisInDegree",
  "componentOfKernel",
  "componentsOfKernel",
  "probCompOfKernel",
  "probCompsOfKernel",
  -- Options
  "Grading", "PreviousGens", "ReturnTargetGrading", "UseMatroidSpeedup", "CoeffField"
}



---------------------
---- maxGrading -----
---------------------
maxGrading = method(Options => {ReturnTargetGrading => false});
maxGrading RingMap := Matrix => opts -> F -> (

  dom := source F;
  codom := target F;
  elimRing := dom ** codom;
  X := vars dom;
  n := numgens dom;
  elimIdeal := ideal(sub(X, elimRing) - sub(F(X), elimRing));
  
  if opts.ReturnTargetGrading then (transpose linealitySpace(gfanHomogeneitySpace(elimIdeal))) else (transpose linealitySpace(gfanHomogeneitySpace(elimIdeal)))_(toList(0..n-1))
  )


TEST ///
A = matrix {{1,1,1,0,0,0,0,0,0}, {0,0,0,1,1,1,0,0,0}, {0,0,0,0,0,0,1,1,1}, {1,0,0,1,0,0,1,0,0}, {0,1,0,0,1,0,0,1,0}};
R = QQ[x_1..x_(numcols A)];
S = QQ[t_1..t_(numrows A)];
F = map(S, R, apply(numcols(A), i -> S_(flatten entries A_i)));
assert(ker(A) == ker(maxGrading(F)));
///



----------------------------
----- trimBasisInDegree ----
----------------------------
trimBasisInDegree = method();
trimBasisInDegree (List, Ring, List, MutableHashTable) := Matrix => (deg, dom, G, basisHash) -> (

  if #G == 0 then (
      return basisHash#deg;
  );

  -- otherwise, we shift G in all possible ways to land in R_deg

  G = apply(G, g -> sub(g, dom));

  L := apply(G, g -> (
          checkDegree := deg - degree(g);
          if basisHash#?checkDegree then (
              g*basisHash#checkDegree
          ) else (
              -- this else condition is only hit when basis(checkDegree,dom) = |0|
              g*basis(checkDegree, dom)
          )
      )
  );
  
  -- stick em all in a matrix
  mat := L#0;
  scan(1..#L-1, i -> mat = mat | L#i);

  -- and collect coefficients.
  (mons, coeffs) := coefficients(mat);

  -- find the independent linear relations
  coeffs = mingens(image sub(coeffs, coefficientRing(dom)));

  -- remove monomials corresponding to pivots
  badMonomials := apply(pivots coeffs, i -> mons_(0,i#0));
  monomialBasis := flatten entries basisHash#deg;
  scan(badMonomials, m -> monomialBasis = delete(m, monomialBasis));

  matrix{monomialBasis}
  )

trimBasisInDegree (List, Ring, MutableHashTable) := Matrix => (deg, dom, basisHash) -> trimBasisInDegree(deg, dom, {}, basisHash)


TEST ///
A = matrix {{1,1,1,0,0,0,0,0,0}, {0,0,0,1,1,1,0,0,0}, {0,0,0,0,0,0,1,1,1}, {1,0,0,1,0,0,1,0,0}, {0,1,0,0,1,0,0,1,0}};
R = QQ[x_1..x_(numcols A)];
S = QQ[t_1..t_(numrows A)];
F = map(S, R, apply(numcols(A), i -> S_(flatten entries A_i)));
dom = newRing(R, Degrees => A);
basisHash = new MutableHashTable from apply(gens(dom), i -> degree(i) => i);
B = basis(2, source F) | basis(3, source F);
lats = unique apply(flatten entries B, i -> degree(sub(i, dom)));
scan(lats, deg -> basisHash#deg = basis(deg, dom));
assert(trimBasisInDegree({2,1,0,1,1},  dom, {x_2*x_4-x_1*x_5, x_3*x_4-x_1*x_6, x_3*x_5-x_2*x_6}, basisHash) == matrix {{x_2*x_3*x_4}});
///



----------------------------
----- componentOfKernel ----
----------------------------
componentOfKernel = method(Options => {PreviousGens => {}});
componentOfKernel (List, Ring, RingMap, Matrix) := List => opts -> (deg, dom, F, monomialBasis) -> (

  -- collect coefficients into a matrix
  (mons, coeffs) := coefficients(F(sub(monomialBasis, source F)));

  -- find the linear relations among coefficients
  K := gens ker sub(coeffs, coefficientRing(dom));

  newGens := flatten entries (monomialBasis * K);

  
  newGens
  )


componentOfKernel (List, Ring, RingMap, MutableHashTable) := List => opts ->  (deg, dom, F, basisHash) -> (

      monomialBasis := if basisHash#?deg then basisHash#deg else trimBasisInDegree(deg, dom, opts.PreviousGens, basisHash);

      componentOfKernel(deg, dom, F, monomialBasis)   
  )


componentOfKernel (List, Ring, RingMap) := List => opts -> (deg, dom, F) -> (


  monomialBasis := basis(deg, dom);

  componentOfKernel(deg, dom, F, monomialBasis)
  )


TEST ///
A = matrix {{1,1,1,0,0,0,0,0,0}, {0,0,0,1,1,1,0,0,0}, {0,0,0,0,0,0,1,1,1}, {1,0,0,1,0,0,1,0,0}, {0,1,0,0,1,0,0,1,0}};
R = QQ[x_1..x_(numcols A)];
S = QQ[t_1..t_(numrows A)];
F = map(S, R, apply(numcols(A), i -> S_(flatten entries A_i)));
dom = newRing(R, Degrees => A);
assert(componentOfKernel({1,1,0,1,1}, dom, F, matrix {{x_1*x_5, x_2*x_4}}) == {x_2*x_4-x_1*x_5});
///


-----------------------------
----- componentsOfKernel ----
-----------------------------
componentsOfKernel = method(Options => {Grading => null, UseMatroidSpeedup => true});
componentsOfKernel (Number, RingMap) := MutableHashTable => opts -> (d, F) -> (

  A := if opts.Grading === null then maxGrading(F) else opts.Grading;
  dom := newRing(source F, Degrees => A);
  basisHash := new MutableHashTable;
  gensHash := new MutableHashTable;

  if (transpose(matrix {toList(numColumns(A) : 1/1)}) % image(transpose sub(A,QQ))) != 0 then (
    print("ERROR: The multigrading does not refine total degree. Try homogenizing or a user-defined multigrading");
    return;
  );

  -- compute the jacobian of F and substitute in random parameter values in a large finite field

  if opts.UseMatroidSpeedup then(

    J := jacobian matrix F;
    J = sub(J, apply(gens target F, t -> t => random(ZZ/nextPrime(100000))));
  );
  
  areThereLinearRelations := false;
  
  -- assumes homogeneous with normal Z-grading
  for i in 1..d do (


    if i == 2 and areThereLinearRelations then print("WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.");
    

    B := sub(basis(i, source F), dom);
    lats := unique apply(flatten entries B, m -> degree m);
    
    G := flatten(values(gensHash));

    for deg in lats do (
      
      basisHash#deg = basis(deg, dom);
      S := findSupportIndices(support sub(basisHash#deg, source F), F);

      if (numcols(basisHash#deg) == 1) and (i > 1) then(

        gensHash#deg = {};
        continue;
      );

      if opts.UseMatroidSpeedup then(


        if rank(J_S) == #S then(

          gensHash#deg = {};
          continue;
          );
        );

      monomialBasis := trimBasisInDegree(deg, dom, G, basisHash);
      gensHash#deg = componentOfKernel(deg, dom, F, monomialBasis);

      if i == 1 and #(gensHash#deg) > 0 then (
        areThereLinearRelations = true;
      );

      );
    );
  
  gensHash
  )

TEST ///
A = matrix {{1,1,1,0,0,0,0,0,0}, {0,0,0,1,1,1,0,0,0}, {0,0,0,0,0,0,1,1,1}, {1,0,0,1,0,0,1,0,0}, {0,1,0,0,1,0,0,1,0}};
R = QQ[x_1..x_(numcols A)];
S = QQ[t_1..t_(numrows A)];
F = map(S, R, apply(numcols(A), i -> S_(flatten entries A_i)));
dom = newRing(R, Degrees => A);
G = componentsOfKernel(2,F);
G = new HashTable from G;
G = delete(null, flatten values(G));
assert(sub(ideal(G),R) == ker F)
///


----------------------------
----- probCompOfKernel ----
----------------------------
probCompOfKernel = method(Options => {PreviousGens => {}});
probCompOfKernel (List, Ring, List, Matrix) := List => opts -> (deg, dom, samplePts, monomialBasis) -> (

  -- collect coefficients into a matrix
  evalBasis := matrix for i from 0 to numcols(monomialBasis)-1 list flatten entries sub(monomialBasis, samplePts_i);

  -- find the linear relations among coefficients
  K := gens ker evalBasis;

  newGens := flatten entries (monomialBasis * K);

  
  newGens
  )


-----------------------------
----- probCompsOfKernel ----
-----------------------------
probCompsOfKernel = method(Options => {Grading => null, UseMatroidSpeedup => true, CoeffField => null});
probCompsOfKernel (Number, RingMap) := MutableHashTable => opts -> (d, F) -> (

  print("warning: computation begun over finite field. resulting polynomials may not lie in the ideal");

  A := if opts.Grading === null then maxGrading(F) else opts.Grading;
  KK := if opts.CoeffField == null then ZZ/nextPrime(1000000) else opts.CoeffField;
  dom := newRing(source F, Degrees => A);
  basisHash := new MutableHashTable;
  gensHash := new MutableHashTable;

  if (transpose(matrix {toList(numColumns(A) : 1/1)}) % image(transpose sub(A,QQ))) != 0 then (
    print("ERROR: The multigrading does not refine total degree. Try homogenizing or a user-defined multigrading");
    return;
  );

  -- compute the jacobian of F and substitute in random parameter values in a large finite field

  if opts.UseMatroidSpeedup then(

    J := jacobian matrix F;
    J = sub(J, apply(gens target F, t -> t => random(KK)));
    );
  
  areThereLinearRelations := false;

  samplePts := {};
  
  -- assumes homogeneous with normal Z-grading
  for i in 1..d do (


    if i == 2 and areThereLinearRelations then print("WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.");
    
    print(concatenate("computing total degree: ", toString(i)));

    B := sub(basis(i, source F), dom);
    lats := unique apply(flatten entries B, m -> degree m);
    scan(lats, deg -> basisHash#deg = basis(deg, dom));
    maxBasisSize := max(apply(values(basisHash), k -> numcols(k)));

    print(concatenate("number of monomials = ", toString(numcols(B))));
    print(concatenate("number of distinct multidegrees = ", toString(#lats)));
    
    -- make list of current generators
    G := flatten(values(gensHash));

    print(concatenate("sampling ", toString(maxBasisSize), " points from the variety"));


    -- sample additional points from the variety if necessary
    if #samplePts <  maxBasisSize then(

        newPts := for l from 0 to (maxBasisSize - #samplePts - 1) list(

          paramVals := apply(gens target F, t -> t => random(KK));
	        
          apply(gens source F, x -> sub(x, dom) => sub(F(x), paramVals))
        );

        samplePts = samplePts | newPts;
      );

    for deg in lats do (
      
      S := findSupportIndices(support sub(basisHash#deg, source F), F);

      if (numcols(basisHash#deg) == 1) and (i > 1) then(

        gensHash#deg = {};
        continue;
        );

      if opts.UseMatroidSpeedup then(


        if rank(J_S) == #S then(

          gensHash#deg = {};
          continue;
          );
        );

      

      monomialBasis := trimBasisInDegree(deg, dom, G, basisHash);
      gensHash#deg = probCompOfKernel(deg, dom, samplePts, monomialBasis);

      if i == 1 and #(gensHash#deg) > 0 then (
        areThereLinearRelations = true;
      );

      );
    );
  
  gensHash
  )

TEST ///
A = matrix {{1,1,1,0,0,0,0,0,0}, {0,0,0,1,1,1,0,0,0}, {0,0,0,0,0,0,1,1,1}, {1,0,0,1,0,0,1,0,0}, {0,1,0,0,1,0,0,1,0}};
R = QQ[x_1..x_(numcols A)];
S = QQ[t_1..t_(numrows A)];
F = map(S, R, apply(numcols(A), i -> S_(flatten entries A_i)));
dom = newRing(R, Degrees => A);
G = componentsOfKernel(2,F);
G = new HashTable from G;
G = delete(null, flatten values(G));
assert(sub(ideal(G),R) == ker F)
///








findSupportIndices = (supp, F) -> (

  apply(supp, s -> position(gens source F, x -> x == s)) 
  )

-- Documentation below

beginDocumentation()

-- template for function documentation
--doc ///
--Key
--Headline
--Usage
--Inputs
--Outputs
--Consequences
--  Item
--Description
--  Text
--  Code
--  Pre
--  Example
--  CannedExample
--Subnodes
--Caveat
--SeeAlso
--///

doc ///
Key
  MultigradedImplicitization
Headline
  Package for levaraging multigradings to solve implicitization problems
Description
  Text
    The MultigradedImplicitization package provides methods for computing the maximal $\mathbb{Z}^k$ grading in which the 
    kernel of a polynomial map $F$ is homogeneous and exploiting it to find generators of $\ker(F)$. This package is
    particularly useful for problems from algebraic statistics which often involve highly structured maps {\tt F} which are
    often naturally homogeneous in a larger multigrading than the standard $\mathbb{Z}$-multigrading given by total degree.
    For more information on this approach see the following:
  Text
    References:

    [1] Cummings, J., & Hollering , B. (2023). Computing Implicitizations of Multi-Graded Polynomial Maps. arXiv preprint arXiv:2311.07678.

    [2] Cummings, J., & Hauenstein, J. (2023). Multi-graded Macaulay Dual Spaces. arXiv preprint arXiv:2310.11587.

    [3] Cummings, J., Hollering, B., & Manon, C. (2024). Invariants for level-1 phylogenetic networks under the cavendar-farris-neyman model. Advances in Applied Mathematics, 153, 102633.
///


doc ///
Key
  maxGrading
  (maxGrading, RingMap)
  [maxGrading, ReturnTargetGrading]
Headline
  computes the maximal $\mathbb{Z}^k$ grading such that $\ker(F)$ is homogeneous
Usage
  maxGrading(F)
Inputs
  F:RingMap
  ReturnTargetGrading => Boolean
    a boolean which encodes whether or not to also return the grading on {\tt target(F)} which induces the grading on $\ker(F)$
Outputs
  :Matrix
    the maximal $\mathbb{Z}^k$ grading in which $\ker(F)$ is homogeneous
--Consequences
--  asd
Description
  Text
    Computes the maximal $\mathbb{Z}^k$ grading such that the $\ker(F)$ is homogeneous. 
    The columns of the output matrix are the degrees of the corresponding variables in {\tt source(F)}.
    For example, the snippet below shows that the maximal grading of a toric ideal
    is exactly the integer matrix which encodes the monomial map parameterizing the ideal.
  Example
    A = matrix {{1,1,1,0,0,0}, {0,0,0,1,1,1}, {1,0,0,1,0,0}, {0,1,0,0,1,0}, {0,0,1,0,0,1}}
    R = QQ[x_(1,1)..x_(2,3)];
    S = QQ[t_1..t_2, s_1..s_3];
    F = map(S, R, {t_1*s_1, t_1*s_2, t_1*s_3, t_2*s_1, t_2*s_2, t_2*s_3})
    maxGrading(F)
  Text
    The option {\tt ReturnTargetGrading} returns a matrix which also gives the corresponding
    grading on the target ring of $F$ which induces the grading on $\ker(F)$. This option is {\tt false}
    by default. 
  Example
    maxGrading(F, ReturnTargetGrading => true)
///


doc ///
Key
  trimBasisInDegree
  (trimBasisInDegree, List, Ring, List, MutableHashTable)
  (trimBasisInDegree, List, Ring, MutableHashTable)
Headline
  Finds a basis for the homogeneous component of a graded ring but removes basis elements which correspond to previously computed generators. 
Usage
  trimBasisInDegree(deg, dom, G, B)
  trimBasisInDegree(deg, dom, B)
Inputs
  deg:List
    the degree of the homogeneous component to compute
  dom:Ring
    a graded ring which is the source of a homogeneous ring map $F$
  G:List
    a list of previously computed generators of $\ker(F)$
  B:MutableHashTable
    a mutable hashtable which contains all bases of homogeneous components which correspond to lower total degrees than {\tt deg}
Outputs
  :Matrix
    A monomial basis for the homogeneous component of degree {\tt deg} of {\tt dom} with any monomials which cannot be involved in new generators of $\ker(F)$ removed.  
--Consequences
--  asd
Description
  Text
    Computes a monomial basis for the homogeneous component of degree {\tt deg} of the graded ring {\tt dom} which is the source of a ring map $F$. 
    Monomials which correspond to previously computed relations which are in {\tt G} are automatically removed since they will not yield new generators
    in $\ker(F)$ when applying @TO2{componentOfKernel, "componentOfKernel"}@ to this basis.  
  Example
    A = matrix {{1,1,1,0,0,0,0,0,0}, {0,0,0,1,1,1,0,0,0}, {0,0,0,0,0,0,1,1,1}, {1,0,0,1,0,0,1,0,0}, {0,1,0,0,1,0,0,1,0}};
    R = QQ[x_1..x_(numcols A)];
    S = QQ[t_1..t_(numrows A)];
    F = map(S, R, apply(numcols(A), i -> S_(flatten entries A_i)));
    dom = newRing(R, Degrees => A);
    basisHash = new MutableHashTable from apply(gens(dom), i -> degree(i) => i);
    B = basis(2, source F) | basis(3, source F);
    lats = unique apply(flatten entries B, i -> degree(sub(i, dom)));
    scan(lats, deg -> basisHash#deg = basis(deg, dom));
    trimBasisInDegree({2,1,0,1,1},  dom, {x_2*x_4-x_1*x_5, x_3*x_4-x_1*x_6, x_3*x_5-x_2*x_6}, basisHash)
  Text
    Observe that after trimming we get a smaller monomial basis for this homogeneous component. The full monomial basis is
  Example
    basis({2,1,0,1,1}, dom)
///


doc ///
Key
  componentsOfKernel
  (componentsOfKernel, Number, RingMap)
  [componentsOfKernel, Grading]
  [componentsOfKernel, UseMatroidSpeedup]
Headline
  Finds all minimal generators up to a given total degree in the kernel of a ring map 
Usage
  componentsOfKernel(d, F)
Inputs
  d:List
    the total degree at which to stop computing generators
  F:RingMap
  Grading => Matrix
    a matrix whose columns give a homogeneous multigrading on $\ker(F)$
  UseMatroidSpeedup => Boolean
    if true, then the jacobian of $F$ is used to detect if it is possible for kernel element to exist in a homogeneous component. If the jacobian does not drop rank, then that component cannot contain kernel generators and is skipped.  
Outputs
  :MutableHashTable
    A mutable hashtable whose keys correspond to all homogeneous components of $\ker(F)$ and values correspond to generators in $\ker(F)$ with those components
--Consequences
--  asd
Description
  Text
    Computes all minimal generators of $\ker(F)$ which are of total degree at most {\tt d}. 
  Example
    A = matrix {{1,1,1,0,0,0}, {0,0,0,1,1,1}, {1,0,0,1,0,0}, {0,1,0,0,1,0}, {0,0,1,0,0,1}}
    R = QQ[x_(1,1)..x_(2,3)];
    S = QQ[t_1..t_2, s_1..s_3];
    F = map(S, R, {t_1*s_1, t_1*s_2, t_1*s_3, t_2*s_1, t_2*s_2, t_2*s_3})
    peek componentsOfKernel(2, F)
  Text
    If a grading in which $\ker(F)$ is homogeneous is already known or a specific grading is desired then the option {\tt Grading} can be used to specify this.
    In this case the columns of the matrix {\tt Grading} are automatically used to grade the source of $F$. 
--  Code
--    todo
--  Pre
--    todo

--  todo
--SeeAlso
--  todo
///


doc ///
Key
  componentOfKernel
  (componentOfKernel, List, Ring, RingMap, Matrix)
  (componentOfKernel, List, Ring, RingMap, MutableHashTable)
  (componentOfKernel, List, Ring, RingMap)
  [componentOfKernel, PreviousGens]
Headline
  Finds all minimal generators of a given degree in the kernel of a ring map 
Usage
  componentOfKernel(deg, dom, F, M)
  componentOfKernel(deg, dom, F, B)
  componentOfKernel(deg, dom, F)
Inputs
  deg:List
    the degree of the homogeneous component to compute
  dom:Ring
    a graded ring which is the source of a homogeneous ring map $F$
  F:RingMap
    a map whose kernel is homogeneous in the grading of {\tt dom}
  M:Matrix
    a monomial basis for the homogeneous component of {\tt deg} with degree {\tt deg}
  B:MutableHashTable
    a mutable hashtable which contains all bases of homogeneous components which correspond to lower total degrees than {\tt deg}
  PreviousGens => List
    a list of generators of the kernel which have lower total degree
Outputs
  :Matrix
    A list of minimal generators for $\ker(F)$ which are in the homogeneous component of degree {\tt deg}
--Consequences
--  asd
Description
  Text
    Computes all minimal generators of $\ker(F)$ which are in the homogeneous component of degree {\tt deg}
  Example
    A = matrix {{1,1,1,0,0,0}, {0,0,0,1,1,1}, {1,0,0,1,0,0}, {0,1,0,0,1,0}, {0,0,1,0,0,1}}
    R = QQ[x_(1,1)..x_(2,3)];
    S = QQ[t_1..t_2, s_1..s_3];
    F = map(S, R, {t_1*s_1, t_1*s_2, t_1*s_3, t_2*s_1, t_2*s_2, t_2*s_3})
    dom = newRing(R, Degrees => A);
    componentOfKernel({1,1,0,1,1}, dom, F)
  Text
      The option {\tt PreviousGens} can be used to specify a set of previously computed generators. 
      In the case that a monomial basis or hash table of monomial bases is not given then @TO2{trimBasisInDegree, "trimBasisInDegree"}@
      will be used to compute a monomial basis and {\tt PreviousGens} will be used to trim this basis. 
--  Code
--    todo
--  Pre
--    todo

--  todo
--SeeAlso
--  todo
///


doc ///
Key
  Grading
Headline
  optional argument 
--Usage
--Inputs
--Outputs
--Consequences
--  Item
Description
  Text
    The option Grading is a @TO2{Matrix,"matrix"}@ that allows one to specify a specific multigrading in which a polynomial map is homogeneous in.
--
--  CannedExample
--Subnodes
--Caveat
--SeeAlso
///


doc ///
Key
  UseMatroidSpeedup
Headline
  optional argument 
--Usage
--Inputs
--Outputs
--Consequences
--  Item
Description
  Text
    The option UseMatroidSpeedup is a boolean that allows one to specify if the matroid given by the jacobian should be used to automatically skip components. This option is true by default.
    If it is set to false, then every homogeneous component will be checked, even if it is impossible for a polynomial with the necessary support to belong to the kernel.
    For very small examples, it may be slightly faster to set this to false. 
--
--  CannedExample
--Subnodes
--Caveat
--SeeAlso
///


doc ///
Key
  ReturnTargetGrading
Headline
  optional argument 
--Usage
--Inputs
--Outputs
--Consequences
--  Item
Description
  Text
    The option ReturnTargetGrading is a @TO2{Boolean,"boolean"}@ which can be used with @TO2{maxGrading, "maxGrading"}@. This option is false by default. If it is set to true
    then the full grading on the elimination ideal of a polynomial map $F$ will be returned instead of only returning the part of the grading which corresponds to the source of $F$.  
--
--  CannedExample
--Subnodes
--Caveat
--SeeAlso
///


doc ///
Key
  PreviousGens
Headline
  optional argument 
--Usage
--Inputs
--Outputs
--Consequences
--  Item
Description
  Text
    The option PreviousGens is a @TO2{List,"list"}@ of polynomials of total degree at most $d-1$ which can be used with @TO2{componentOfKernel, "componentOfKernel"}@
    to trim the monomial basis for that multidegree. 
--
--  CannedExample
--Subnodes
--Caveat
--SeeAlso
///