Re: how to keep up to a certain degree of a multivariate polynomial?

[Simon King]

Hi!

On 2013-06-04, Sam math <hes...@gmail.com> wrote:
> How do I do this for a multivariate polynomial? It says O(.) is not defined...
>
> R.<x,y> = PolynomialRing(QQ)
>
> f = x^3*y^3 + x^2 * y^4 + x*y + x + y + 1
>
> How can I chop this polynomial up to a certain degree of x and y? I.e. I want to keep up until the second degree of x only (regardless of y).

I guess it depends on your application. If you *absolutely* need O(...),
and if your *really* want to bound only the x-degree and keep y untouched,
then you could define f as a *univariate* polynomial in x over a
polynomial ring in y:

sage: f = x^3*y^3 + x^2 * y^4 + x*y + x + y + 1
sage: f
y^3*x^3 + y^4*x^2 + (y + 1)*x + y + 1
sage: f + O(x^2)
y + 1 + (y + 1)*x + O(x^2)

Be warned, however, that arithmetic in a polynomial ring over a
polynomial ring will be dramatically slower than in a multi-variate
polynomial ring:

sage: R2.<x,y> = QQ[]
sage: g = x^3*y^3 + x^2 * y^4 + x*y + x + y + 1
sage: type(f)
sage.rings.polynomial.polynomial_element.Polynomial_generic_dense
sage: type(g)
sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular
sage: %timeit f*f
10000 loops, best of 3: 128 us per loop
sage: %timeit g*g
100000 loops, best of 3: 3.57 us per loop
sage: %timeit f+f
100000 loops, best of 3: 3.15 us per loop
sage: %timeit g+g
1000000 loops, best of 3: 1.47 us per loop

Best regards,
Simon

[john_perry_usm]

Here's a different approach, which is more efficient, but poses its own challenges:

  sage: I = R.ideal(x^2)
  sage: Q = R.quo(I)
  sage: f = Q(x^3*y^3 + x^2*y^4 + x*y + x + y + 1)
  sage: f
  xbar*ybar + xbar + ybar + 1

So, the variables look different, and with reason. But:

  sage: %timeit g*g
  625 loops, best of 3: 3.4 µs per loop
  sage: %timeit f*f
  625 loops, best of 3: 30.5 µs per loop
  sage: %timeit g+g
  625 loops, best of 3: 2.39 µs per loop
  sage: %timeit f+f
  625 loops, best of 3: 27.2 µs per loop

Multiplication is faster than Simon's solution, but addition is (curiously) much slower.

regards
john perry

On Monday, June 3, 2013 7:12:34 PM UTC-5, Sam math wrote:

I have a multivariate polynomial and want to keep only up to a certain degree. I already know how to do this for the univariate case.

For 1 variable, I'd do:

R.<x> = PolynomialRing(QQ)

f = x^4 + x^2 + x^3 + x + 1

f = f + O(x^3)

print f

#output would be 1 + x + x^2... which is what I want.

How do I do this for a multivariate polynomial? It says O(.) is not defined...

R.<x,y> = PolynomialRing(QQ)

f = x^3*y^3 + x^2 * y^4 + x*y + x + y + 1

How can I chop this polynomial up to a certain degree of x and y? I.e. I want to keep up until the second degree of x only (regardless of y).

I know one way I could go about this is to just redefine the polynomial by looping through the first few coefficients, but I'm looking for a more efficient/easier way.

Thanks!

[Stephen Montgomery-Smith]

> On Monday, June 3, 2013 7:12:34 PM UTC-5, Sam math wrote:
>
> I have a multivariate polynomial and want to keep only up to a
> certain degree. I already know how to do this for the univariate case.
>
> For 1 variable, I'd do:
>
> R.<x> = PolynomialRing(QQ)
>
> f = x^4 + x^2 + x^3 + x + 1
>
> f = f + O(x^3)
>
> print f
>
> #output would be 1 + x + x^2... which is what I want.
>
> How do I do this for a multivariate polynomial? It says O(.) is not
> defined...
>
> R.<x,y> = PolynomialRing(QQ)
>
> f = x^3*y^3 + x^2 * y^4 + x*y + x + y + 1
>
> How can I chop this polynomial up to a certain degree of x and y?
> I.e. I want to keep up until the second degree of x only (regardless
> of y).
>
> I know one way I could go about this is to just redefine the
> polynomial by looping through the first few coefficients, but I'm
> looking for a more efficient/easier way.
>
> Thanks!

I think what Sam Math wants to do is this: he has two multivariate
polynomials f and g, lets say in x and y. What he wants to do is to
multiply f*g, and then throw away all the terms of degree higher than,
say, x^100, or even all terms of degree higher than x^100 and y^200.

He doesn't necessarily want a beautiful solution. What he wants is an
efficient solution.

Maybe "looping through the first few coefficients" is the best way to
go. But any better solutions would be greatly appreciated.

[leif]

sage: f
x^3*y^3 + x^2*y^4 + x*y + x + y + 1
sage: f.truncate(x,3)
x^2*y^4 + x*y + x + y + 1
sage: f.truncate(x,3).truncate(y,3)

x*y + x + y + 1

-leif

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