如何在Julia的AbstractAlgebra.jl中定义可计算行列式的多项式矩阵？

Hey there! Let's break down how to fix this issue so you can compute the determinant of your polynomial matrix correctly.

The Problem with Your Current Code #

Your code creates a nested Julia array (Vector{Vector{elem_type(S)}}), but AbstractAlgebra.jl's det function only works with its native ring matrix type (MatrixElem), not base Julia arrays. That's why you can't call det(M) on your current setup. Also, keep in mind: determinants are only defined for square matrices—so if your matrix has 8 rows and n columns (with n≠8), you'll need to adjust the dimensions first (either make it n×n or 8×8 depending on your goal).

The Solution: Use AbstractAlgebra's matrix Constructor #

You can easily convert your nested array into a valid ring matrix using the matrix function from AbstractAlgebra. Here are two straightforward ways to do this:

1. Directly Construct the Matrix from a Nested Array

If you already have your nested array (like the one you wrote), just pass it to matrix along with the target ring S:

using AbstractAlgebra

n = 15
T, x = PolynomialRing(ZZ, n, "x")
S, y = PolynomialRing(T, n , "y")

# First create your nested array (adjust dimensions to square if needed)
# Example: n×n square matrix (since det requires square)
arr = [[x[i]*y[j] for i=1:n] for j=1:n]

# Convert to AbstractAlgebra matrix over ring S
M = matrix(S, arr)

2. Construct the Matrix Directly from a Flat Element List

Alternatively, you can build the matrix by passing a flat list of elements (ordered column-wise by default, but the nested array method is more intuitive for row-based construction):

# Build an 8×8 square matrix (if you want 8 rows/columns instead of n×n)
M = matrix(S, 8, 8, [x[i]*y[j] for j=1:8 for i=1:8])

Calculate the Determinant #

Once M is a valid MatrixElem instance, you can call det(M) and get a polynomial in ring S as expected:

# Compute determinant
det_M = det(M)

# Verify the type (should be an element of S)
println(typeof(det_M))  # Output: elem_type(S)

Example with Small n #

Let's test with n=2 to see the result clearly:

n = 2
T, x = PolynomialRing(ZZ, n, "x")
S, y = PolynomialRing(T, n , "y")

arr = [[x[i]*y[j] for i=1:n] for j=1:n]
M = matrix(S, arr)
det_M = det(M)

println(det_M)  # Output: x1*y1*x2*y2 - x1*y2*x2*y1 (which simplifies to 0, since this is a rank-1 matrix)

Key Notes #

• Always ensure your matrix is square before calling det—non-square matrices don't have a determinant.
• AbstractAlgebra's MatrixElem type is designed to handle algebraic operations (like determinant calculation) over rings, which base Julia arrays don't support natively.

内容的提问来源于stack exchange，提问作者skywalker