polynomialmatricestest.nb

单模多项式矩阵的分解算法

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Cell["Diophantine Equations ", "Section",
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The degree of a polynomial vector is the maximun degree of its \
entries.
The function DiophantineSolve finds two polynomial matrices X, Y (if any) of \
minimal column degree that solve the equation a.X+b.Y==c, where a, b, c, are \
polynomial matrices with the same number of rows\
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It is also possible to compute ALL the minimal column degree \
solutions of the diophantine equation: it is sufficient to specify one more \
variable that will be used to describe the free variables of the result\
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Cell["\<\
For every real or complex value of the t[i]'s this is a minimal \
column degree solution of the diophantine equation.
The following is an example of a diophantine equation which has no \
solution:\
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Cell["Some other useful functions", "Section",
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Cell["\<\
A matrix is said to be column reduced if the matrix of the leading \
coefficients has full column rank. A polynomial matrix can always be put in \
column reduced form using only elementary operations on the columns; \
PolynomialColumnReduce computes the unimodular matrix that represents these \
elementary operations. \
\>", "Text",
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    \(\(m = {{x\^2 - 1, x\^3}, {1, x - 1}, {2  x, 
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To put the matrix m in column reduced form it is sufficient to \
postmultiply it by the unimodular matrix just computed\
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PolynomialRowReduce[m] computes a unimodular matrix U such that U.m \
is row reduced.
The function Rank[m] returns the rank of matrix m and EliminateDep[m] \
eliminates the dependent rows of m:\
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Cell["\<\
The function PolyExtendedGCD is the polynomial version of \
ExtendedGCD:\
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Cell["?PolyExtendedGCD", "Input",
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Cell[BoxData[
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null) and returns a list {g,{r,s}}, where g is a monic GCD of a and b, and \
r,s s.t. g=ar+bs."\)], "Print"],

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Cell[BoxData[
    \(1 + x\)], "Output"],

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Cell["\<\
PolynomialDegree[m,x] returns the degree of a matrix (or vector) \
m\
\>", "Text",
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Cell[BoxData[
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Cell[TextData[{
  "ExtPolQ[list,x] yields True if all the entries of \"list\" are polynomials \
in x or in ",
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