20140808, 11:57  #1 
Nov 2003
16100_{8} Posts 
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. 
20140808, 12:37  #2 
(loop (#_fork))
Feb 2006
Cambridge, England
2×29×109 Posts 
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; nonlinear 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. 
20140808, 13:08  #3  
Nov 2003
2^{6}·113 Posts 
Quote:
Quote:
Quote:
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] 

20140808, 13:17  #4 
Nov 2003
7232_{10} Posts 
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 20140808 at 13:18 Reason: mismatched quotes 
20140808, 13:26  #5  
Nov 2003
2^{6}×113 Posts 
Quote:
I originally put it in the Misc. Math. subforum for a reason. 

20140808, 14:26  #6  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
2·3·7·241 Posts 
Quote:
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 corecipients.) Paul Added in edit: first I need to discover an identity. Not provingas easy as expected. Last fiddled with by xilman on 20140808 at 14:28 

20140808, 15:11  #7 
Romulan Interpreter
Jun 2011
Thailand
2·5·887 Posts 
I also don't get it, if I don't have some Englishrelated 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 20140808 at 15:12 
20140808, 22:21  #8  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2^{2}·1,433 Posts 
Quote:
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. 

20140809, 05:46  #9 
Romulan Interpreter
Jun 2011
Thailand
2·5·887 Posts 
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 20140809 at 05:47 
20140809, 12:42  #10  
"Forget I exist"
Jul 2009
Dumbassville
8,369 Posts 
Quote:
Last fiddled with by science_man_88 on 20140809 at 13:16 

20140810, 08:25  #11  
May 2004
New York City
10146_{8} Posts 
Quote:
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 cofinite 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 cofinite. 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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