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SectionalMatrix

sectional matrix

Syntax
SectionalMatrix(I: IDEAL): type MAT
SectionalMatrix(PmodI: RING): type MAT
SectionalMatrix(I: IDEAL, bound:INT): type MAT
SectionalMatrix(PmodI: RING, bound:INT): type MAT

Description
The definition of Hilbert function was extended in "Borel Sets and Sectional Matrices" to the bivariate function encoding the Hilbert functions of the generic hyperplane sections: the sectional matrix of I (homogenous ideal) or PmodI .

The second argument makes a matrix with bound columns. The default value is reg(I) (since the rest of the matrix is obtained by the Persistence Theorem).

See articles Bigatti, Robbiano, "Borel Sets and Sectional Matrices", and Bigatti, Palezzato, Torielli, "Sectional Matrices" (work in progress).

Example
/**/ use P ::= QQ[x,y,z];
/**/ I := ideal(x^4 -x*y^3, x*y -z^2, x*z^2 -y^3);
/**/ SectionalMatrix(I);
matrix(ZZ,
 [[0, 0, 1, 1, 1, 1, 1, 1],
  [0, 0, 1, 3, 5, 6, 7, 8],
  [0, 0, 1, 4, 10, 18, 26, 34]])
/**/ SectionalMatrix(P/I);
matrix(ZZ,
 [[1, 1, 0, 0, 0, 0, 0, 0],
  [1, 2, 2, 1, 0, 0, 0, 0],
  [1, 3, 5, 6, 5, 3, 2, 2]])

/**/ SectionalMatrix(P/I, 10);
matrix(ZZ,
 [[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
  [1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0],
  [1, 3, 5, 6, 5, 3, 2, 2, 2, 2, 2]])