padiclib.spad line 261 [edit on github]
In this package K is a finite field, R is a ring of univariate polynomials over K, and F is a monogenic algebra over R. We require that F is monogenic, i.e. that F = K[x, y]/(f(x, y)), because the integral basis algorithm used will factor the polynomial f(x, y). The package provides a function to compute the integral closure of R in the quotient field of F as well as a function to compute a "local integral basis" at a specific prime.
integralBasis() returns a record [basis, basisDen, basisInv] containing information regarding the integral closure of R in the quotient field of the framed algebra F. F is a framed algebra with R-module basis w1, w2, ..., wn. If 'basis' is the matrix (aij, i = 1..n, j = 1..n), then the ith element of the integral basis is vi = (1/basisDen) * sum(aij * wj, j = 1..n)ith row of 'basis' contains the coordinates of the ith basis vector. Similarly, the ith row of the matrix 'basisInv' contains the coordinates of wiv1, ..., vn: if 'basisInv' is the matrix (bij, i = 1..n, j = 1..n), then wi = sum(bij * vj, j = 1..n)
integralBasis(p) returns a record [basis, basisDen, basisInv] containing information regarding the local integral closure of R at the prime p in the quotient field of the framed algebra F. F is a framed algebra with R-module basis w1, w2, ..., wn. If 'basis' is the matrix (aij, i = 1..n, j = 1..n), then the ith element of the local integral basis is vi = (1/basisDen) * sum(aij * wj, j = 1..n)ith row of 'basis' contains the coordinates of the ith basis vector. Similarly, the ith row of the matrix 'basisInv' contains the coordinates of wiv1, ..., vn: if 'basisInv' is the matrix (bij, i = 1..n, j = 1..n), then wi = sum(bij * vj, j = 1..n)
reducedDiscriminant(up) is undocumented