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Old 2014-08-08, 11:57   #1
R.D. Silverman
 
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Default Polynomial Algebra

OK. So you (the audience) think you understand polynomial algebra
at the high school level.

I toss out the following problem:

Prove or Disprove:

Let f(x) and g(x) be any irreducible polynomials over Z.

Proposition: The range sets of f(x) and g(x) are cofinite.

If false, tell us under what conditions the proposition might be true.
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Old 2014-08-08, 12:37   #2
fivemack
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I think I'm missing something - should there be a connection between f and g? Should the polynomials be homogeneous in two variables?

The range sets of x^3+2 and x^3+3 are both coinfinite and their union is also coinfinite; non-linear polynomials grow fast.

On the other hand this caused me to look at http://en.wikipedia.org/wiki/15_and_290_theorems which are really nice theorems.
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Old 2014-08-08, 13:08   #3
R.D. Silverman
 
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Quote:
Originally Posted by fivemack View Post
I think I'm missing something - should there be a connection between f and g?
No.

Quote:
Should the polynomials be homogeneous in two variables?
Not as proposed.

Quote:
The range sets of x^3+2 and x^3+3 are both coinfinite and their union is also coinfinite; non-linear polynomials grow fast.
The point of the post is to try to get people to think and perhaps
to get them to learn some algebra (A forlorn hope, I know). Nothing more.

On the other hand this caused me to look at http://en.wikipedia.org/wiki/15_and_290_theorems which are really nice theorems.[/QUOTE]
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Old 2014-08-08, 13:17   #4
R.D. Silverman
 
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Quote:
Originally Posted by R.D. Silverman View Post
No

Not as proposed. .




The point of the post is to try to get people to think and perhaps
to get them to learn some algebra (A forlorn hope, I know). Nothing more.

Here's another question/food for thought:

When is the intersection of the range sets of two different irreducible
polynomials (in one variable) infinite? When might it be finite?

I only toss this (and the other problem) out in an attempt to get people
to actually think about polynomial behavior, instead of spewing nonsense.
(And perhaps to get people to learn a little set theory to understand
'cofinite')

It is a Socratic teaching attempt. (probably futile, but who knows?)

Last fiddled with by R.D. Silverman on 2014-08-08 at 13:18 Reason: mismatched quotes
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Old 2014-08-08, 13:26   #5
R.D. Silverman
 
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Quote:
Originally Posted by R.D. Silverman View Post
The point of the post is to try to get people to think and perhaps
to get them to learn some algebra (A forlorn hope, I know). Nothing more.

Here's another question/food for thought:

When is the intersection of the range sets of two different irreducible
polynomials (in one variable) infinite? When might it be finite?

I only toss this (and the other problem) out in an attempt to get people
to actually think about polynomial behavior, instead of spewing nonsense.
(And perhaps to get people to learn a little set theory to understand
'cofinite')

It is a Socratic teaching attempt. (probably futile, but who knows?)
BTW. Why was my first post in this thread moved here?
I originally put it in the Misc. Math. subforum for a reason.
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Old 2014-08-08, 14:26   #6
xilman
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Quote:
Originally Posted by R.D. Silverman View Post
BTW. Why was my first post in this thread moved here? I originally put it in the Misc. Math. subforum for a reason.
Sounds like a misunderstanding to me. A joint misunderstanding, that is. The Misc. Math is explicitly and exclusively for crank mathematics (though I may also have misunderstood) and your posting gives every indication of not being the output of a crank of that variety. The mover clearly misunderstood your intention.

Given that I could also be misunderstanding the situation on multiple levels I will not move the thread back. I'm chickening out and leaving it to be discussed between you and the mod in question. (In a further display of cowardice, I won't post the mover's identity here but will send a PM each of you as co-recipients.)

Paul

Added in edit: first I need to discover an identity. Not provingas easy as expected.

Last fiddled with by xilman on 2014-08-08 at 14:28
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Old 2014-08-08, 15:11   #7
LaurV
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I also don't get it, if I don't have some English-related misunderstanding, if f(x) is a polynomial over Z, its range set is cofinite only if the degree of the polynomial is one. Otherwise, it will leave out an infinite number of values, in fact, it will take less values than it will leave out. Or I am totally in the weeds? And why we need g(x) here? (the second polynomial?).

[And don't put the topic back in misc math! Whatever the outcome is from here, I feel I will learn something, and I honestly like this face of RDS better than the other one So let the topic live!]

Last fiddled with by LaurV on 2014-08-08 at 15:12
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Old 2014-08-08, 22:21   #8
henryzz
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Quote:
Originally Posted by LaurV View Post
I also don't get it, if I don't have some English-related misunderstanding, if f(x) is a polynomial over Z, its range set is cofinite only if the degree of the polynomial is one. Otherwise, it will leave out an infinite number of values, in fact, it will take less values than it will leave out. Or I am totally in the weeds? And why we need g(x) here? (the second polynomial?).

[And don't put the topic back in misc math! Whatever the outcome is from here, I feel I will learn something, and I honestly like this face of RDS better than the other one So let the topic live!]
You give conditions for the range set of f(x) to be cofinite. Cofinite with what? It is the two range sets that are supposed to be cofinite.
Either the range sets are both equal or the question means that one of the range sets is a cofinite subset of the other and not both ways.

Hopefully my before bed post isn't gibberish. I am slightly surprised that I didn't come across cofiniteness in my maths degree. I suppose a lot of the time when we looked at set theory we looked at mainly the bits we needed.
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Old 2014-08-09, 05:46   #9
LaurV
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Thanks for the explanation.

So, in fact, RDS's "proposition" states that "there is only a finite set of elements in union(f,g) which is not in intersection(f,g)". Is this what is asked? Just to make it clear what the question is. In which case, the solution is quite simple. Evey problem is simple to solve if the right questions are asked

(And one who gives a solution should use spoilers )

Last fiddled with by LaurV on 2014-08-09 at 05:47
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Old 2014-08-09, 12:42   #10
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Quote:
Originally Posted by LaurV View Post
Thanks for the explanation.

So, in fact, RDS's "proposition" states that "there is only a finite set of elements in union(f,g) which is not in intersection(f,g)". Is this what is asked? Just to make it clear what the question is. In which case, the solution is quite simple. Evey problem is simple to solve if the right questions are asked

(And one who gives a solution should use spoilers )
If I did my research correctly what is being proposed is that f(x) and g(x) are irreducible polynomials in Z, and that their images on one another are cofinite( aka the complement of there image on the other is finite) that's my understanding ( though admittedly the only thing I found that actually said range set was something starting with I related to sets, but wasn't image) edit: prove this true or false, if false list the conditions for it to be true. edit2: doh injective function on wikipedia is where I caught mention of it

Last fiddled with by science_man_88 on 2014-08-09 at 13:16
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Old 2014-08-10, 08:25   #11
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Quote:
Originally Posted by R.D. Silverman View Post
Prove or Disprove:

Let f(x) and g(x) be any irreducible polynomials over Z.

Proposition: The range sets of f(x) and g(x) are cofinite.

If false, tell us under what conditions the proposition might be true.
(I think the following is what was intended.)

Polynomials over Z are necessarily of finite degree, hence take
on infinitely many values in their range, whether the polynomial
is irreducible or not.

Two sets, such as the respective ranges, are co-finite under teo
possible meanings: either (a) they are either both finite or both
infinite, or (b) each is a subset of the other whose complement
in the other is finite.

(a) is true for any range(f) and range(g), since both are infinite;

(b) is true since if two sets are subsets of each other they must
be equal, hence easily co-finite.

Change Z to Z/nZ and everything becomes finite, so the proposition
becomes true by finiteness instead of infiniteness.

(Of course, this analysis ignores the irreducibility condition and
makes the problem trivial, which may not have been what was
intended by the OP.)


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