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jetMatrix -- construct the matrix of k-jets evaluated at a given point.

Synopsis

Usage:

jetMatrix(A,k,pt)
Inputs:
- A, a matrix,
- k, an integer,
- pt, a matrix,
Outputs:
- a matrix,

Description

The function constructs the matrix of k-jets evaluated at the point pt for the polarized toric variety associated to the set of lattice points A.

i1 : A=latticePoints(convexHull(matrix{{0,0,2},{0,2,0}}))

o1 = {0, | 0 |, | 0 |, | 1 |, | 1 |, | 2 |}
         | 1 |  | 2 |  | 0 |  | 1 |  | 0 |

o1 : List

i2 : pt=matrix{{1},{1}}

o2 = | 1 |
     | 1 |

              2       1
o2 : Matrix ZZ  <-- ZZ

i3 : jetMatrix(A,2,pt)

o3 = | 1 1 1 1 1 1 |
     | 0 0 0 1 1 2 |
     | 0 0 0 0 0 2 |
     | 0 0 0 0 1 0 |
     | 0 1 2 0 1 0 |
     | 0 0 2 0 0 0 |

              6       6
o3 : Matrix ZZ  <-- ZZ

If no point is provided the matrix of k-jets is provided as a matrix over a polynomial ring.

i4 : A=latticePoints(convexHull(matrix{{0,0,2},{0,2,0}}))

o4 = {0, | 0 |, | 0 |, | 1 |, | 1 |, | 2 |}
         | 1 |  | 2 |  | 0 |  | 1 |  | 0 |

o4 : List

i5 : jetMatrix(A,2)

o5 = | 1 x_1 x_1^2 x_0 x_0x_1 x_0^2 |
     | 0 0   0     1   x_1    2x_0  |
     | 0 0   0     0   0      2     |
     | 0 0   0     0   1      0     |
     | 0 1   2x_1  0   x_0    0     |
     | 0 0   2     0   0      0     |

                        6                 6
o5 : Matrix (ZZ[x ..x ])  <-- (ZZ[x ..x ])
                 0   1             0   1

See also

isJetSpanned -- checks if the polarized toric variety associated to a set of lattice points is k-jet spanned at a given point.

Ways to use jetMatrix:

jetMatrix(List,ZZ)
jetMatrix(List,ZZ,Matrix)
jetMatrix(Matrix,ZZ)
jetMatrix(Matrix,ZZ,Matrix)

For the programmer

The object jetMatrix is a method function.