|
RegularChains[FastArithmeticTools][NormalizePolynomialDim0] - normalize a polynomial w.r.t a 0-dim regular chain
|
|
Calling Sequence
|
|
NormalizePolynomialDim0(f, rc, R)
|
|
|
Parameters
|
|
|
R
|
-
|
a polynomial ring
|
|
rc
|
-
|
a regular chain of R
|
|
f
|
-
|
polynomial of R
|
|
|
|
|
|
Description
|
|
|
•
|
The command NormalizePolynomialDim0 returns a normalized form of f w.r.t. rc, that is, a polynomial  which is associated to f modulo rc, such that  is normalized w.r.t. rc.
|
|
•
|
rc is zero-dimensional regular chain, and f together with rc forms a zero-dimensional regular chain.
|
|
|
|
Examples
|
|
|
>
|
|
|
>
|
|
|
>
|
|
|
>
|
|
|
>
|
![vars := [y, x]](/support/helpjp/helpview.aspx?si=6448/file06494/math128.png)
|
|
>
|
|
We consider two bivariate polynomials and want to compute their common solutions
|
>
|
|
|
>
|
|
We first compute their subresultant chain using FFT techniques
|
>
|
|
|
| (1) |
We deduce their resultants
|
>
|
|
|
| (2) |
We observe below that no root of r2 cancels the leading coefficients of f1 or f2. Hence, any roots of r2 can be extended into a solution of the system by a GCD computation.
|
>
|
|
|
| (3) |
We define the regular chain consisting of r2
|
>
|
![rc := Chain([r2], Empty(R), R)](/support/helpjp/helpview.aspx?si=6448/file06494/math175.png)
|
|
| (4) |
We compute the GCD of f1 and f2 modulo r2
|
>
|
|
|
![g2 := [[x*y^2+x*y+2*x+y^2+y+2+x^3, regular_chain], [x*y^2+x*y+2*x+y^2+y+2+x^3, regular_chain]]](/support/helpjp/helpview.aspx?si=6448/file06494/math187.png)
| (5) |
We normalize this GCD w.r.t. r2 which leads to a simpler expression with one as leading coefficient
|
>
|
![NormalizePolynomialDim0(g2[1][1], g2[1][2], R)](/support/helpjp/helpview.aspx?si=6448/file06494/math193.png)
|
|
| (6) |
|
|
Download Help Document
Was this information helpful?
