up previous next
FrobeniusMat    --    matrix of the Frobenius Map


Syntax 
FrobeniusMat(I: IDEAL): MAT
FrobeniusMat(I: IDEAL, QB: LIST): MAT

Description
The Frobenius map on R/I sends f --> f^q. This function computes the matrix of the Frobenius Map in R/I where I is a zero-dimensional ideal. The second uses the given quotient basis, otherwise the Macaulay basis is used.

Example 
/**/  use P ::= ZZ/(5)[x,y,z], Lex;
/**/  I := ideal(y^2-x*z, z^2-x^2*y, x+y+z-1);

/**/  FrobeniusMat(I);
matrix( /*RingWithID(3, "FFp(5)")*/
 [[1, 0, 0, 0, 0, 0],
  [0, -2, 2, -2, -1, 0],
  [0, 1, 2, -1, 2, 0],
  [0, 2, 2, -1, -1, 0],
  [0, 1, 1, -1, -2, 1],
  [0, 0, 0, 0, 0, 0]])

/**/  FrobeniusMat(I, [z^4, z^3, z^2, z, y, 1]);
matrix( /*RingWithID(3, "FFp(5)")*/
 [[-2, -1, 1, 1, 1, 0],
  [-1, -1, 2, 2, 0, 0],
  [2, -1, 2, 1, 0, 0],
  [-1, -2, 2, -2, 0, 0],
  [0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 1]])

See Also