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Polynomial remainder sequence and approximate GCD
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Let P
1
and P
2
be polynomials, univariate or multivariate, and let (P
1
, P
2
, P
3
,…, P
i
,…) be a polynomial remainder sequence. Let A
i
and B
i
(i = 3, 4,…) be polynomials such that A
i
P
1
+ B
i
P
2
= P
i
, deg(A
i
) < deg(P
2
) - deg(P
i
), deg(B
i
) < deg(P
1
) - deg(P
i
), where the degree is for the main variable. We derive relations such as C
i
P
1
= -B
i+1
P
i
+ B
i
P
i+1
and C
i
P
2
= A
i+1
P
i
- A
i
P
i+1
, where C
i
is independent of the main variable. Using these relations, we discuss approximate common divisors calculated by polynomial remainder sequence.
Title: Polynomial remainder sequence and approximate GCD
Description:
Let P
1
and P
2
be polynomials, univariate or multivariate, and let (P
1
, P
2
, P
3
,…, P
i
,…) be a polynomial remainder sequence.
Let A
i
and B
i
(i = 3, 4,…) be polynomials such that A
i
P
1
+ B
i
P
2
= P
i
, deg(A
i
) < deg(P
2
) - deg(P
i
), deg(B
i
) < deg(P
1
) - deg(P
i
), where the degree is for the main variable.
We derive relations such as C
i
P
1
= -B
i+1
P
i
+ B
i
P
i+1
and C
i
P
2
= A
i+1
P
i
- A
i
P
i+1
, where C
i
is independent of the main variable.
Using these relations, we discuss approximate common divisors calculated by polynomial remainder sequence.
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