Theorems about ideals

fivemack · 2012-01-20, 21:42

Suppose I have an ideal of Q[x,y,z,t] generated by f1 and f2, each homogenous of degree 2.

Is there a theorem that, if a polynomial G of degree d is in the ideal, it can be written as q1*f1 + q2*f2 with deg(q1), deg(q2) bounded? I'm not at all sure even where to start looking for such a thing.

ccorn · 2012-01-20, 22:11

Quote:

Originally Posted by fivemack

Suppose I have an ideal of Q[x,y,z,t] generated by f1 and f2, each homogenous of degree 2.

Is there a theorem that, if a polynomial G of degree d is in the ideal, it can be written as q1*f1 + q2*f2 with deg(q1), deg(q2) bounded? I'm not at all sure even where to start looking for such a thing.

What about using a Groebner basis f1', f2', expressing G with respect to that base (giving q1', q2'), and then transforming back?

fivemack · 2012-01-20, 22:29

Quote:

Originally Posted by ccorn

What about using a Groebner basis f1', f2', expressing G with respect to that base (giving q1', q2'), and then transforming back?

That's what the software I'm using at the moment (sage) is doing; it works perfectly well and determines q1 and q2 happily, but I don't know how to get a degree bound out of the process. The problem is that sage leaks memory at a spectacular rate, and I want to do tests as to whether a fixed G is in the ideals generated by a few billion different pairs f1, f2.

ccorn · 2012-01-20, 23:33

Quote:

Originally Posted by fivemack

Suppose I have an ideal of Q[x,y,z,t] generated by f1 and f2, each homogenous of degree 2.

Is there a theorem that, if a polynomial G of degree d is in the ideal, it can be written as q1*f1 + q2*f2 with deg(q1), deg(q2) bounded? I'm not at all sure even where to start looking for such a thing.

From googling for "degree bounds on Bezout polynomials", I have found this article.

Zeta-Flux · 2012-01-21, 00:09

fivemack,

That should follow directly from the fact that $f_1$ and $f_2$ are homogeneous.

Given $G$ , write it as $G=\sum_{i} G_i$ , where $G_i$ is the homogeneous part of $G$ in degree $i$ .

If $G=q_1*f_1 + q_2*f2$ and $q_{i,j}$ is the homogeneous part of $q_i$ in degree $j$ then $G_i=q_{1,i-2}f_1 + q_{2,i-2}f_2$ . (To see this, just look at all of the possible terms of a given degree.)

In particular, you can choose the degrees on the $q$ 's to be bounded by $\deg(G)-2$ .

ccorn · 2012-01-21, 08:06

Quote:

Originally Posted by ccorn

From googling for "degree bounds on Bezout polynomials", I have found this article.

This mainly considers zero-dimensional zero loci. Your problem has too many variables for that, sorry. Yet another interesting article is this one which seems to cover your problem.

ccorn · 2012-01-21, 08:51

Quote:

Originally Posted by ccorn

This mainly considers zero-dimensional zero loci. Your problem has too many variables for that, sorry. Yet another interesting article is this one which seems to cover your problem.

In particular, theorem 1.2 (p. 104) of the above article by Andersson mostly confirms Zeta-Flux's claim.

ccorn · 2012-01-21, 10:21

Quote:

Originally Posted by Zeta-Flux

fivemack,

That should follow directly from the fact that $f_1$ and $f_2$ are homogeneous.

Given $G$ , write it as $G=\sum_{i} G_i$ , where $G_i$ is the homogeneous part of $G$ in degree $i$ .

If $G=q_1*f_1 + q_2*f2$ and $q_{i,j}$ is the homogeneous part of $q_i$ in degree $j$ then $G_i=q_{1,i-2}f_1 + q_{2,i-2}f_2$ . (To see this, just look at all of the possible terms of a given degree.)

In particular, you can choose the degrees on the $q$ 's to be bounded by $\deg(G)-2$ .

I see. Even if higher-degree parts of q1, q2 would lead to a complete cancellation in the linear combination, those parts do not influence lower-degree parts of the linear combination, so they can as well be left out.

fivemack · 2012-01-21, 11:37

Thanks very much. I feel silly now, I should have been able to construct zeta-flux's argument on my own ...

Zeta-Flux · 2012-01-21, 15:30

Don't feel bad, this is one of those things that just appears easy in hindsight. My area of expertise is ring theory, and quite a few of my papers specifically work with homogeneous ideals to construct examples and counter-examples. So I'm just very familiar with this kind of argument.

ccorn · 2012-01-22, 11:01

Thanks to Zeta-Flux, the original question has been answered.

However---Fivemack: Is your G a homogenized polynomial?

If so, assume t is the projection variable. Then G and G*t^k (for some natural number k) are equivalent in the sense that their affine (unhomogenized) counterparts are equal.
And it may well happen that G is not in the ideal generated by f1, f2 whereas
G*t^k is (from some k onward).

Example:

f1(x,y,z,t) = y^2 + t^2
f2(x,y,z,t) = xy - xt + yt
G(x,y,z,t) = xy - y^2 + xt

All these are degree 2, but you need degree-1 factors to combine f1, f2 to G*t:

q1(x,y,z,t) = x
q2(x,y,z,t) = -y

I note this rather for myself in order to keep in mind why homogenizing does not magically ease all membership questions.

2012-01-20, 21:42	#1
fivemack (loop (#_fork)) Feb 2006 Cambridge, England 1100011101111₂ Posts	Theorems about ideals Suppose I have an ideal of Q[x,y,z,t] generated by f1 and f2, each homogenous of degree 2. Is there a theorem that, if a polynomial G of degree d is in the ideal, it can be written as q1f1 + q2f2 with deg(q1), deg(q2) bounded? I'm not at all sure even where to start looking for such a thing.

2012-01-21, 00:09	#5
Zeta-Flux May 2003 7·13·17 Posts	fivemack, That should follow directly from the fact that $f_1$ and $f_2$ are homogeneous. Given $G$ , write it as $G=\sum_{i} G_i$ , where $G_i$ is the homogeneous part of $G$ in degree $i$ . If $G=q_1f_1 + q_2f2$ and $q_{i,j}$ is the homogeneous part of $q_i$ in degree $j$ then $G_i=q_{1,i-2}f_1 + q_{2,i-2}f_2$ . (To see this, just look at all of the possible terms of a given degree.) In particular, you can choose the degrees on the $q$ 's to be bounded by $\deg(G)-2$ . Last fiddled with by Zeta-Flux on 2012-01-21 at 00:09

2012-01-21, 15:30	#10
Zeta-Flux May 2003 1547₁₀ Posts	Don't feel bad, this is one of those things that just appears easy in hindsight. My area of expertise is ring theory, and quite a few of my papers specifically work with homogeneous ideals to construct examples and counter-examples. So I'm just very familiar with this kind of argument. Last fiddled with by Zeta-Flux on 2012-01-21 at 15:31

2012-01-22, 11:01	#11
ccorn Apr 2010 2×3×5² Posts	Thanks to Zeta-Flux, the original question has been answered. However---Fivemack: Is your G a homogenized polynomial? If so, assume t is the projection variable. Then G and Gt^k (for some natural number k) are equivalent in the sense that their affine (unhomogenized) counterparts are equal. And it may well happen that G is not in the ideal generated by f1, f2 whereas Gt^k is (from some k onward). Example: f1(x,y,z,t) = y^2 + t^2 f2(x,y,z,t) = xy - xt + yt G(x,y,z,t) = xy - y^2 + xt All these are degree 2, but you need degree-1 factors to combine f1, f2 to Gt: q1(x,y,z,t) = x q2(x,y,z,t) = -y I note this rather for myself in order to keep in mind why homogenizing does not magically ease all membership questions. Last fiddled with by ccorn on 2012-01-22 at 11:08*

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
NFS: Sieving the norm over ideals vs. integers	paul0	Factoring	2	2015-01-19 11:52
Factorization of Ideals in Number Field, Looking for Reference	jinydu	Abstract Algebra & Algebraic Number Theory	5	2014-07-30 11:24
Mersenne theorems question	ShiningArcanine	Math	21	2012-04-27 01:38
Counting ideals during singleton removal	fivemack	Factoring	5	2007-12-30 11:17
new theorems about primes	sghodeif	Miscellaneous Math	18	2006-07-13 18:24

2012-01-21, 11:37	#9
fivemack (loop (#_fork)) Feb 2006 Cambridge, England 13·491 Posts	Thanks very much. I feel silly now, I should have been able to construct zeta-flux's argument on my own ...