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Calling Sequence
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QRationalCanonicalForm[1](F, q, n)
QRationalCanonicalForm[2](F, q, n)
QRationalCanonicalForm[3](F, q, n)
QRationalCanonicalForm[4](F, q, n)
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Parameters
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F
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rational function of n
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q
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name used as the parameter q, usually q
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n
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variable
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Description
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Let F be a rational function of n over a field K of characteristic 0, q is a nonzero element of K which is not a root of unity. The QRationalCanonicalForm[i](F,q,n) command constructs the th rational canonical form for F, .
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If QRationalCanonicalForm is called without an index, the first q-rational canonical form is constructed.
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The output is a sequence of 5 elements , called , where z is an element of K, and are monic polynomials over K such that:
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1.
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, .
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2.
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for all integers .
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4.
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, .
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Note: Q is the automorphism of K(n) defined by .
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The five-tuple that satisfies the four conditions is a strict q-rational normal form for F. The rational function and are called the kernel and the shell of the , respectively.
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Let be any qRNF of a rational function F. Then the degrees of the polynomials r and s are unique, and have minimal possible values in the sense that if where p, q are polynomials in n, and G is a rational function of n, then and .
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Additionally, if then is minimal; if then is minimal; if then is minimal, and under this condition, is minimal; if then is minimal, and under this condition, is minimal.
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Examples
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Check the result from QRationalCanonicalForm[2].
Condition 1 is satisfied.
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Condition 2 is satisfied.
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Condition 3 is satisfied.
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Condition 4 is satisfied.
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Degrees of the kernel:
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The degree of v1 is minimal:
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The degree of u2 is minimal:
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For , the degree of the shell is minimal:
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References
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Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Efficient Representations of (q-)Hypergeometric Terms and the Assignment Problem." Submitted.
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Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Rational Canonical Forms and Efficient Representations of Hypergeometric Terms." Proc. ISSAC'2003, pp. 7-14. 2003.
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