Return the polynomial of the sums of permanental minors of A.
INPUT:
The polynomial of the sums of permanental minors is
where \(p_i(A)\) is the \(i\)-th permanental minor of \(A\) (that can also be obtained through the method permanental_minor() via A.permanental_minor(i)).
The algorithm implemented by that function has been developed by P. Butera and M. Pernici, see [ButPer]. Its complexity is \(O(2^n m^2 n)\) where \(m\) and \(n\) are the number of rows and columns of \(A\). Moreover, if \(A\) is a banded matrix with width \(w\), that is \(A_{ij}=0\) for \(|i - j| > w\) and \(w < n/2\), then the complexity of the algorithm is \(O(4^w (w+1) n^2)\).
INPUT:
EXAMPLES:
sage: from sage.matrix.matrix_misc import permanental_minor_polynomial
sage: m = matrix([[1,1],[1,2]])
sage: permanental_minor_polynomial(m)
3*t^2 + 5*t + 1
sage: permanental_minor_polynomial(m, permanent_only=True)
3
sage: permanental_minor_polynomial(m, prec=2)
5*t + 1
sage: M = MatrixSpace(ZZ,4,4)
sage: A = M([1,0,1,0,1,0,1,0,1,0,10,10,1,0,1,1])
sage: permanental_minor_polynomial(A)
84*t^3 + 114*t^2 + 28*t + 1
sage: [A.permanental_minor(i) for i in range(5)]
[1, 28, 114, 84, 0]
An example over \(\QQ\):
sage: M = MatrixSpace(QQ,2,2)
sage: A = M([1/5,2/7,3/2,4/5])
sage: permanental_minor_polynomial(A, True)
103/175
An example with polynomial coefficients:
sage: R.<a> = PolynomialRing(ZZ)
sage: A = MatrixSpace(R,2)([[a,1], [a,a+1]])
sage: permanental_minor_polynomial(A, True)
a^2 + 2*a
A usage of the var argument:
sage: m = matrix(ZZ,4,[0,1,2,3,1,2,3,0,2,3,0,1,3,0,1,2])
sage: permanental_minor_polynomial(m, var='x')
164*x^4 + 384*x^3 + 172*x^2 + 24*x + 1
ALGORITHM:
The permanent \(perm(A)\) of a \(n \times n\) matrix \(A\) is the coefficient of the \(x_1 x_2 \ldots x_n\) monomial in
\[\prod_{i=1}^n \left( \sum_{j=1}^n A_{ij} x_j \right)\]Evaluating this product one can neglect \(x_i^2\), that is \(x_i\) can be considered to be nilpotent of order \(2\).
To formalize this procedure, consider the algebra \(R = K[\eta_1, \eta_2, \ldots, \eta_n]\) where the \(\eta_i\) are commuting, nilpotent of order \(2\) (i.e. \(\eta_i^2 = 0\)). Formally it is the quotient ring of the polynomial ring in \(\eta_1, \eta_2, \ldots, \eta_n\) quotiented by the ideal generated by the \(\eta_i^2\).
We will mostly consider the ring \(R[t]\) of polynomials over \(R\). We denote a generic element of \(R[t]\) by \(p(\eta_1, \ldots, \eta_n)\) or \(p(\eta_{i_1}, \ldots, \eta_{i_k})\) if we want to emphasize that some monomials in the \(\eta_i\) are missing.
Introduce an “integration” operation \(\langle p \rangle\) over \(R\) and \(R[t]\) consisting in the sum of the coefficients of the non-vanishing monomials in \(\eta_i\) (i.e. the result of setting all variables \(\eta_i\) to \(1\)). Let us emphasize that this is not a morphism of algebras as \(\langle \eta_1 \rangle^2 = 1\) while \(\langle \eta_1^2 \rangle = 0\)!
Let us consider an example of computation. Let \(p_1 = 1 + t \eta_1 + t \eta_2\) and \(p_2 = 1 + t \eta_1 + t \eta_3\). Then
\[p_1 p_2 = 1 + 2t \eta_1 + t (\eta_2 + \eta_3) + t^2 (\eta_1 \eta_2 + \eta_1 \eta_3 + \eta_2 \eta_3)\]and
\[\langle p_1 p_2 \rangle = 1 + 4t + 3t^2\]In this formalism, the permanent is just
\[perm(A) = \langle \prod_{i=1}^n \sum_{j=1}^n A_{ij} \eta_j \rangle\]A useful property of \(\langle . \rangle\) which makes this algorithm efficient for band matrices is the following: let \(p_1(\eta_1, \ldots, \eta_n)\) and \(p_2(\eta_j, \ldots, \eta_n)\) be polynomials in \(R[t]\) where \(j \ge 1\). Then one has
\[\langle p_1(\eta_1, \ldots, \eta_n) p_2 \rangle = \langle p_1(1, \ldots, 1, \eta_j, \ldots, \eta_n) p_2 \rangle\]where \(\eta_1,..,\eta_{j-1}\) are replaced by \(1\) in \(p_1\). Informally, we can “integrate” these variables before performing the product. More generally, if a monomial \(\eta_i\) is missing in one of the terms of a product of two terms, then it can be integrated in the other term.
Now let us consider an \(m \times n\) matrix with \(m \leq n\). The sum of permanental `k`-minors of `A` is
\[perm(A, k) = \sum_{r,c} perm(A_{r,c})\]where the sum is over the \(k\)-subsets \(r\) of rows and \(k\)-subsets \(c\) of columns and \(A_{r,c}\) is the submatrix obtained from \(A\) by keeping only the rows \(r\) and columns \(c\). Of course \(perm(A, \min(m,n)) = perm(A)\) and note that \(perm(A,1)\) is just the sum of all entries of the matrix.
The generating function of these sums of permanental minors is
\[g(t) = \left\langle \prod_{i=1}^m \left(1 + t \sum_{j=1}^n A_{ij} \eta_j\right) \right\rangle\]In fact the \(t^k\) coefficient of \(g(t)\) corresponds to choosing \(k\) rows of \(A\); \(\eta_i\) is associated to the i-th column; nilpotency avoids having twice the same column in a product of \(A\)‘s.
For more details, see the article [ButPer].
From a technical point of view, the product in \(K[\eta_1, \ldots, \eta_n][t]\) is implemented as a subroutine in prm_mul(). The indices of the rows and columns actually start at \(0\), so the variables are \(\eta_0, \ldots, \eta_{n-1}\). Polynomials are represented in dictionary form: to a variable \(\eta_i\) is associated the key \(2^i\) (or in Python 1 << i). The keys associated to products are obtained by considering the development in base \(2\): to the monomial \(\eta_{i_1} \ldots \eta_{i_k}\) is associated the key \(2^{i_1} + \ldots + 2^{i_k}\). So the product \(\eta_1 \eta_2\) corresponds to the key \(6 = (110)_2\) while \(\eta_0 \eta_3\) has key \(9 = (1001)_2\). In particular all operations on monomials are implemented via bitwise operations on the keys.
REFERENCES:
| [ButPer] | P. Butera and M. Pernici “Sums of permanental minors using Grassmann algebra”, Arxiv 1406.5337 |
Return the product of p1 and p2, putting free variables in mask_free to \(1\).
This function is mainly use as a subroutine of permanental_minor_polynomial().
INPUT:
EXAMPLES:
sage: from sage.matrix.matrix_misc import prm_mul
sage: t = polygen(ZZ, 't')
sage: p1 = {0: 1, 1: t, 4: t}
sage: p2 = {0: 1, 1: t, 2: t}
sage: prm_mul(p1, p2, 1, None)
{0: 2*t + 1, 2: t^2 + t, 4: t^2 + t, 6: t^2}
This function computes a weak Popov form of a matrix over a rational function field \(k(x)\), for \(k\) a field.
INPUT:
- \(M\) - matrix
- \(ascend\) - if True, rows of output matrix \(W\) are sorted so degree (= the maximum of the degrees of the elements in the row) increases monotonically, and otherwise degrees decrease.
OUTPUT:
A 3-tuple \((W,N,d)\) consisting of two matrices over \(k(x)\) and a list of integers:
\(N\) is invertible over \(k(x)\). These matrices satisfy the relation \(N*M = W\).
EXAMPLES:
The routine expects matrices over the rational function field, but other examples below show how one can provide matrices over the ring of polynomials (whose quotient field is the rational function field).
sage: R.<t> = GF(3)['t']
sage: K = FractionField(R)
sage: import sage.matrix.matrix_misc
sage: sage.matrix.matrix_misc.row_reduced_form(matrix([[(t-1)^2/t],[(t-1)]]))
(
[ 0] [ t 2*t + 1]
[(2*t + 1)/t], [ 1 2], [-Infinity, 0]
)
NOTES:
See docstring for row_reduced_form method of matrices for more information.