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Math Puzzle #3

Blogs > mieda
Post a Reply
1 2 3 Next All
mieda
Profile Blog Joined February 2010
United States85 Posts
Last Edited: 2010-09-12 22:32:52
September 10 2010 20:12 GMT
#1
Let f,g be polynomials with rational coefficients. Suppose f(Q) = g(Q) (i.e. the sets of values of f and g on the rationals are the same). Then f(x) = g(ax + b) for some rational constants a,b.

Remark. It generalizes to number fields with real embeddings.

Note that the converse is trivial.

Solution

The idea is that (mod the full details) there's a fixed discrete subgroup L of Q such that for any integer x, the rational solutions, viewed in y, of g(y) = f(x) are all in L. To extract the behavior of y as x varies, and compare the coefficients at the same time, i.e. the growth rates, you can consider expressions of the form g(y') - g(y) = f(x+1) - f(x) where y' is the corresponding rational number such that g(y') = f(x+1). By mean value theorem (don't quite need this, since f,g are polynomials so one can potentially do all the computations by algebra) the expression extracts out y' - y in terms of f'(x) and g'(y) roughly, and this gives a good comparison of y as x varies. This is the idea, and I do some computations to show that this works.

Also, simple valuation calculations (plug in this, plug in that rational number, compare the p-adic valuations) don't really seem to get anywhere, as I give plenty of counter examples and point out the problems to the claims of one of the posters below.

*
Assault_1
Profile Joined April 2009
Canada1950 Posts
September 10 2010 20:34 GMT
#2
so whats the question?
mieda
Profile Blog Joined February 2010
United States85 Posts
September 10 2010 20:43 GMT
#3
On September 11 2010 05:34 Assault_1 wrote:
so whats the question?


Proving it.
Piege
Profile Joined October 2009
United Kingdom128 Posts
September 10 2010 20:52 GMT
#4
Would a non-math major/non-math enthusiast be able to solve this?
Never_V_ -> Fin
category
Profile Joined July 2009
United States85 Posts
September 10 2010 20:53 GMT
#5
the sets f(Q) and g(Q) are not counting multiplicity right?
category
Profile Joined July 2009
United States85 Posts
September 10 2010 20:54 GMT
#6
On September 11 2010 05:52 Piege wrote:
Would a non-math major/non-math enthusiast be able to solve this?


I don't know the solution yet but I suspect the answer to your question is no.
mieda
Profile Blog Joined February 2010
United States85 Posts
Last Edited: 2010-09-10 20:56:54
September 10 2010 20:55 GMT
#7
On September 11 2010 05:53 category wrote:
the sets f(Q) and g(Q) are not counting multiplicity right?


They're just sets, not really any structures on them (for ex. ramifications). For example for the functio f(x) = x^2 the set f(Q) is just the set of all squares of rational numbers. We don't care that f(x) = 0 ramifies (and elsewhere unramifies). It's just a naive set.

I wonder if that's what you meant. So yes, I think we're not counting multiplicities here, whatever you mean by that in this context. f(Q) = {y : y = f(a) for some rational a}.
category
Profile Joined July 2009
United States85 Posts
September 10 2010 20:59 GMT
#8
lol. I don't know about ramifications. i just meant, eg in the f(x^2) case, does it matter that 1 is obtained twice? but it sounds like no.

i guess for the functions f=x^2 and g=x^4, f(Q) will be larger than g(Q), because some rationals are the square of another rational but are not a fourth power?
mieda
Profile Blog Joined February 2010
United States85 Posts
Last Edited: 2010-09-10 21:01:28
September 10 2010 21:00 GMT
#9
On September 11 2010 05:59 category wrote:
lol. I don't know about ramifications. i just meant, eg in the f(x^2) case, does it matter that 1 is obtained twice? but it sounds like no.


Right, just count it once. So for example {1,1} = {1}.

i guess for the functions f=x^2 and g=x^4, f(Q) will be larger than g(Q), because some rationals are the square of another rational but are not a fourth power?


That's right.
Assault_1
Profile Joined April 2009
Canada1950 Posts
September 10 2010 21:04 GMT
#10
Then f(x) = g(ax + b) for some rational constants a,b.


whats the domain of x, rational or real?
mieda
Profile Blog Joined February 2010
United States85 Posts
Last Edited: 2010-09-10 21:12:02
September 10 2010 21:06 GMT
#11
On September 11 2010 06:04 Assault_1 wrote:
Show nested quote +
Then f(x) = g(ax + b) for some rational constants a,b.


whats the domain of x, rational or real?


any field containing the rationals. It really doesn't matter, since the condition is only imposed on f(Q) and g(Q). You don't need to "see" bigger fields containing the rationals. Any such field is infinite anyway, so if the condition f(x) = g(ax + b) holds for evaluating x on all rationals then it's identical as polynomials (in the ring Q[x]) so evaluation of x on any field containing Q likewise.
category
Profile Joined July 2009
United States85 Posts
September 10 2010 21:09 GMT
#12
On September 11 2010 06:04 Assault_1 wrote:
Show nested quote +
Then f(x) = g(ax + b) for some rational constants a,b.


whats the domain of x, rational or real?


i would say, think of it as being the rationals.

but yeah, i am finding this quite hard, though I find the result very interesting.
sputnik.theory
Profile Blog Joined July 2009
Poland449 Posts
September 10 2010 21:16 GMT
#13
I've noticed that threads like this pop up during the school year...
ITT: TL does your math homework for you
“On the night of the murder I was at home, asleep. The characters in my dream can vouch for me.”
mieda
Profile Blog Joined February 2010
United States85 Posts
September 10 2010 21:16 GMT
#14
On September 11 2010 06:16 sputnik.theory wrote:
I've noticed that threads like this pop up during the school year...
ITT: TL does your math homework for you


Rest assured, I've already solved this. It's not a homework.
Seth_
Profile Blog Joined July 2010
Belgium184 Posts
September 10 2010 21:18 GMT
#15
May I (a non-math major) ask for an example of an f(x) and g(x) polynomial for which this is true.
Fly[DCT]
Profile Joined September 2010
Canada38 Posts
Last Edited: 2010-09-10 21:25:16
September 10 2010 21:18 GMT
#16
On September 11 2010 06:04 Assault_1 wrote:
Show nested quote +
Then f(x) = g(ax + b) for some rational constants a,b.


whats the domain of x, rational or real?


I am guessing the real or complex is fine. Once we can prove this for the real it automatically implies that it's also true for the complex.

On September 11 2010 06:18 Seth_ wrote:
May I (a non-math major) ask for an example of an f(x) and g(x) polynomial for which this is true.


Sure. f(x) = x^2. g(x) = (2x)^2.
lalalalala
Muirhead
Profile Blog Joined October 2007
United States556 Posts
September 10 2010 21:25 GMT
#17
f(Q) clearly equals g(Q) if f(x)=g(ax+b)

Thus WLOG f and g have integer coefficients that are relatively prime and 0 as their constant terms.

Suppose f(x)=a_nx^n+...a_1*x.

Suppose g(x)=b_nx^n+...+b_1*x.

Let p^r be a prime power dividing a_1 but not b_1, if such a prime power exists. Then f(x) will always be divisible by p either less than or equal to 0 times or more than r times, while g(p) is divisible by p at least once and at most r times.

So this means that f(Q)=g(Q) then a_1= plus/minus b_1

One can continue inductively through all the coefficients
starleague.mit.edu
Muirhead
Profile Blog Joined October 2007
United States556 Posts
September 10 2010 21:26 GMT
#18
Or something like that... don't have time to check everything but should be close
starleague.mit.edu
mieda
Profile Blog Joined February 2010
United States85 Posts
Last Edited: 2010-09-10 21:48:28
September 10 2010 21:29 GMT
#19
On September 11 2010 06:25 Muirhead wrote:
f(Q) clearly equals g(Q) if f(x)=g(ax+b)

Thus WLOG f and g have integer coefficients that are relatively prime and 0 as their constant terms.

Suppose f(x)=a_nx^n+...a_1*x.

Suppose g(x)=b_nx^n+...+b_1*x.

Let p^r be a prime power dividing a_1 but not b_1, if such a prime power exists. Then f(x) will always be divisible by p either less than or equal to 0 times or more than r times, while g(p) is divisible by p at least once and at most r times.

So this means that f(Q)=g(Q) then a_1= plus/minus b_1

One can continue inductively through all the coefficients


Need a lot more details, and I'm not convinced of this argument either. You're saying WLOG assume 0 as their constant terms? And then you're assuming deg f = deg g (this is true, but you didn't mention how to prove this at all). And then there's the next paragraph which doesn't seem to make much sense to me.

Enjoy your Friday anyway!
naptiem
Profile Joined July 2009
United States21 Posts
Last Edited: 2010-09-10 21:33:06
September 10 2010 21:29 GMT
#20
Here's an attempt for any field, F:

+ Show Spoiler +

Since f(F)=g(F), g(F) includes f(x) and there must be an element s of F where g(s)=f(x).

Rewrite s in the form ax+b for some a,b in F:

If x = 0, let b = s and a be any number in F.
Then ax+b = a 0 + s = s.

If x is not 0, let a = x^-1, the multiplicative inverse of x in F, and b = s - 1.
Then ax+b = (x^-1) x + (s - 1) = 1 + (s - 1) = s.

In both cases, ax+b = s and g(ax+b) = g(s) = f(x).


Edit: Never mind, I think I misread the problem.
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