construct four rational canonical forms of a rational function

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SumTools[Hypergeometric][RationalCanonicalForm] - construct four rational canonical forms of a rational function

	Calling Sequence
	RationalCanonicalForm[1](F, n) RationalCanonicalForm[2](F, n) RationalCanonicalForm[3](F, n) RationalCanonicalForm[4](F, n)

Parameters

F	-	rational function of n
n	-	variable

Description

•	Let F be a rational function of n over a field K of characteristic 0. The RationalCanonicalForm[i](F,n) calling sequence constructs the ith rational canonical forms for F, .

If the RationalCanonicalForm command is called without an index, the first rational canonical form is constructed.

•	The output is a sequence of 5 elements , called , where z is an element of K, and are monic polynomials over K such that:

F = z*r*E(u/v)*v/(s*u) .

2.	for all integers k.

, .

Note: E is the automorphism of K(n) defined by .

•	The five-tuple that satisfies the three conditions is a strict rational normal form for F. The rational functions and are called the kernel and the shell of an , respectively.

•	Let be any RNF 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 .

If then is minimal.

If then is minimal, and under this condition, is minimal.

Examples

(1)

(2)

F := n*(n+2)*(n-4+2^(1/2))*(n-3+2^(1/2))*(n+2+2^(1/2))*(n+11+2^(1/2))/((n-3)*(n-2)^2*(n+6)*(n+12)*(n-1+2^(1/2))*(n+1+2^(1/2)))

(3)

z1, r1, s1, u1, v1 := 1, (n-4+2^(1/2))*(n-3+2^(1/2)), (n-3)*(n+6)*(n+12), (n+1+2^(1/2))^2*(n-1)^2*(n-2)^2*(n+1)*n*(n+10+2^(1/2))*(n+9+2^(1/2))*(n+8+2^(1/2))*(n+7+2^(1/2))*(n+6+2^(1/2))*(n+5+2^(1/2))*(n+4+2^(1/2))*(n+3+2^(1/2))*(n+2+2^(1/2))*(n+2^(1/2))*(n-1+2^(1/2)), 1

(4)

z2, r2, s2, u2, v2 := 1, (n+2+2^(1/2))*(n+11+2^(1/2)), (n-3)*(n-2)^2, 1, (n-2+2^(1/2))^2*(n-3+2^(1/2))^2*(n+5)^2*(n+4)^2*(n+3)^2*(n+2)^2*(n+2^(1/2))*(n-1+2^(1/2))*(n-4+2^(1/2))*(n+11)*(n+10)*(n+9)*(n+8)*(n+7)*(n+6)*(n+1)*n

(5)

(6)

(7)

Check the result from RationalCanonicalForm[1].

Condition 1 is satisfied.

evalb(F = normal(z1*r1*subs(n = n+1, u1/v1)*v1/(s1*u1)))

(8)

Condition 2 is satisfied.

(9)

Condition 3 is satisfied.

(10)

Degrees of the kernel:

(11)

(12)

The degree of v1 is minimal.

(13)

The degree of u2 is minimal.

(14)

For , the degree of the shell is minimal.

(15)

	See Also
	evalb, LREtools[dispersion], subs, SumTools[Hypergeometric], SumTools[Hypergeometric][EfficientRepresentation], SumTools[Hypergeometric][MultiplicativeDecomposition], SumTools[Hypergeometric][PolynomialNormalForm], SumTools[Hypergeometric][SumDecomposition]

References

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.

Abramov, S.A., and Petkovsek, M. "Canonical representations of hypergeometric terms." Proc. FPSAC'2001, pp. 1-10. 2001.

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