I recently learned from the book "Mathematical Omnibus" by Fuchs and Tabachnikov that problem A1 on the 1969 Putnam turned out to be much harder than intended because the people setting the exam had an incorrect "solution" in mind. Only 1% of the test takers scored 8/10 or higher.
The crux of the matter was this: Let f be a polynomial function from R^2 to R, i.e., a polynomial in two real variables x and y with real coefficients. Is it possible for the range of f to be the open interval (0, infinity)?
Although there is a short, elementary solution, it is surprisingly tricky to come up with it. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
On Aug 27, 3:25 pm, tc...@lsa.umich.edu wrote (in part):
> The crux of the matter was this: Let f be a polynomial > function from R^2 to R, i.e., a polynomial in two real > variables x and y with real coefficients. Is it possible > for the range of f to be the open interval (0, infinity)?
I tried to come up with an example and couldn't think of something after a few minutes, so I started to think of ways to disprove it. What about a compactness argument? Assuming the polynomial isn't constant (otherwise, can't get this range), any such example would be an example of a continuous function defined on the Riemann sphere whose range is the non-compact subset (0, oo] of the extended real line. (Haven't thought this over very carefully and, no, I haven't looked at a solution.)
> I recently learned from the book "Mathematical Omnibus" by Fuchs and > Tabachnikov that problem A1 on the 1969 Putnam turned out to be much > harder than intended because the people setting the exam had an > incorrect "solution" in mind. Only 1% of the test takers scored 8/10 > or higher.
> The crux of the matter was this: Let f be a polynomial function from > R^2 to R, i.e., a polynomial in two real variables x and y with real > coefficients. Is it possible for the range of f to be the open interval > (0, infinity)?
> Although there is a short, elementary solution, it is surprisingly > tricky to come up with it.
I`m not an expert, but I try anyway. R^2 is both open and closed, and because a continuous mapping (polynomial) maps open sets to open sets and closed to closed sets, there is no example where a closed set R^2 gets mapped to a non-closed set (0,oo).
tc...@lsa.umich.edu wrote: > The crux of the matter was this: Let f be a polynomial function from > R^2 to R, i.e., a polynomial in two real variables x and y with real > coefficients. Is it possible for the range of f to be the open interval > (0, infinity)?
Perhaps some traction can be attained by considering the polynomials f_theta(r) = f(r cos theta, r sin theta). Each is a polynomial in r that attains its minimum at solutions of f_theta'(r)=0, and the degrees of the f_theta are bounded by deg f.
Perhaps the set {(r,theta) : f_theta'(r)=0} is compact, or at least well-enough behaved that we can show that f attains its minimum there. But I have no proof.
> > I recently learned from the book "Mathematical Omnibus" by Fuchs and > > Tabachnikov that problem A1 on the 1969 Putnam turned out to be much > > harder than intended because the people setting the exam had an > > incorrect "solution" in mind. Only 1% of the test takers scored 8/10 > > or higher.
> > The crux of the matter was this: Let f be a polynomial function from > > R^2 to R, i.e., a polynomial in two real variables x and y with real > > coefficients. Is it possible for the range of f to be the open interval > > (0, infinity)?
> > Although there is a short, elementary solution, it is surprisingly > > tricky to come up with it.
> I`m not an expert, but I try anyway. R^2 is both open and closed, and > because a continuous mapping (polynomial) maps open sets to open sets > and closed to closed sets, there is no example where a closed set R^2 > gets mapped to a non-closed set (0,oo).
If your argument were correct, the image of *every* continuous real function defined on R^2 would be both open and closed and hence, since R is connected, surjective. There are, though, examples of continuous functions R^2 --> R, even polynomial ones, which are not surjective.
> > > I recently learned from the book "Mathematical Omnibus" by Fuchs and > > > Tabachnikov that problem A1 on the 1969 Putnam turned out to be much > > > harder than intended because the people setting the exam had an > > > incorrect "solution" in mind. Only 1% of the test takers scored 8/10 > > > or higher.
> > > The crux of the matter was this: Let f be a polynomial function from > > > R^2 to R, i.e., a polynomial in two real variables x and y with real > > > coefficients. Is it possible for the range of f to be the open interval > > > (0, infinity)?
> > > Although there is a short, elementary solution, it is surprisingly > > > tricky to come up with it.
> > I`m not an expert, but I try anyway. R^2 is both open and closed, and > > because a continuous mapping (polynomial) maps open sets to open sets > > and closed to closed sets, there is no example where a closed set R^2 > > gets mapped to a non-closed set (0,oo).
> If your argument were correct, the image of *every* > continuous real function defined on R^2 would be both open and > closed and hence, since R is connected, surjective. > There are, though, examples of continuous functions > R^2 --> R, even polynomial ones, which are not surjective.
> <mariano.suarezalva...@gmail.com> wrote: > > On Aug 27, 7:07 pm, Gc <Gcut...@hotmail.com> wrote:
> > > On 27 elo, 23:25, tc...@lsa.umich.edu wrote:
> > > > I recently learned from the book "Mathematical Omnibus" by Fuchs and > > > > Tabachnikov that problem A1 on the 1969 Putnam turned out to be much > > > > harder than intended because the people setting the exam had an > > > > incorrect "solution" in mind. Only 1% of the test takers scored 8/10 > > > > or higher.
> > > > The crux of the matter was this: Let f be a polynomial function from > > > > R^2 to R, i.e., a polynomial in two real variables x and y with real > > > > coefficients. Is it possible for the range of f to be the open interval > > > > (0, infinity)?
> > > > Although there is a short, elementary solution, it is surprisingly > > > > tricky to come up with it.
> > > I`m not an expert, but I try anyway. R^2 is both open and closed, and > > > because a continuous mapping (polynomial) maps open sets to open sets > > > and closed to closed sets, there is no example where a closed set R^2 > > > gets mapped to a non-closed set (0,oo).
> > If your argument were correct, the image of *every* > > continuous real function defined on R^2 would be both open and > > closed and hence, since R is connected, surjective. > > There are, though, examples of continuous functions > > R^2 --> R, even polynomial ones, which are not surjective.
> OK. Can you give me an example?
An example of what? Of a non-surjective function from R^2 to R?
> > I recently learned from the book "Mathematical Omnibus" by Fuchs and > > Tabachnikov that problem A1 on the 1969 Putnam turned out to be much > > harder than intended because the people setting the exam had an > > incorrect "solution" in mind. Only 1% of the test takers scored 8/10 > > or higher.
> > The crux of the matter was this: Let f be a polynomial function from > > R^2 to R, i.e., a polynomial in two real variables x and y with real > > coefficients. Is it possible for the range of f to be the open interval > > (0, infinity)?
> > Although there is a short, elementary solution, it is surprisingly > > tricky to come up with it.
> I`m not an expert, but I try anyway. R^2 is both open and closed, and > because a continuous mapping (polynomial) maps open sets to open sets > and closed to closed sets, there is no example where a closed set R^2 > gets mapped to a non-closed set (0,oo).
I`ll correct that continuous functions has a preimage open for open sets and closed for closed sets. Not the same as "maps open sets to open sets etc..".So my argument fails.
>> > > I recently learned from the book "Mathematical Omnibus" by Fuchs and >> > > Tabachnikov that problem A1 on the 1969 Putnam turned out to be much >> > > harder than intended because the people setting the exam had an >> > > incorrect "solution" in mind. Only 1% of the test takers scored 8/10 >> > > or higher.
>> > > The crux of the matter was this: Let f be a polynomial function from >> > > R^2 to R, i.e., a polynomial in two real variables x and y with real >> > > coefficients. Is it possible for the range of f to be the open interval >> > > (0, infinity)?
>> > > Although there is a short, elementary solution, it is surprisingly >> > > tricky to come up with it.
>> > I`m not an expert, but I try anyway. R^2 is both open and closed, and >> > because a continuous mapping (polynomial) maps open sets to open sets >> > and closed to closed sets, there is no example where a closed set R^2 >> > gets mapped to a non-closed set (0,oo).
>> If your argument were correct, the image of *every* >> continuous real function defined on R^2 would be both open and >> closed and hence, since R is connected, surjective. >> There are, though, examples of continuous functions >> R^2 --> R, even polynomial ones, which are not surjective. > OK. Can you give me an example?
An example of what? The mapping (x,y) |-> x^2 + y^2 is not surjective, though it doesn't satisfy the requested property.
<mariano.suarezalva...@gmail.com> wrote: > On Aug 27, 7:43 pm, Gc <Gcut...@hotmail.com> wrote:
> > On 28 elo, 01:38, Mariano Suárez-Alvarez
> > <mariano.suarezalva...@gmail.com> wrote: > > > On Aug 27, 7:07 pm, Gc <Gcut...@hotmail.com> wrote:
> > > > On 27 elo, 23:25, tc...@lsa.umich.edu wrote:
> > > > > I recently learned from the book "Mathematical Omnibus" by Fuchs and > > > > > Tabachnikov that problem A1 on the 1969 Putnam turned out to be much > > > > > harder than intended because the people setting the exam had an > > > > > incorrect "solution" in mind. Only 1% of the test takers scored 8/10 > > > > > or higher.
> > > > > The crux of the matter was this: Let f be a polynomial function from > > > > > R^2 to R, i.e., a polynomial in two real variables x and y with real > > > > > coefficients. Is it possible for the range of f to be the open interval > > > > > (0, infinity)?
> > > > > Although there is a short, elementary solution, it is surprisingly > > > > > tricky to come up with it.
> > > > I`m not an expert, but I try anyway. R^2 is both open and closed, and > > > > because a continuous mapping (polynomial) maps open sets to open sets > > > > and closed to closed sets, there is no example where a closed set R^2 > > > > gets mapped to a non-closed set (0,oo).
> > > If your argument were correct, the image of *every* > > > continuous real function defined on R^2 would be both open and > > > closed and hence, since R is connected, surjective. > > > There are, though, examples of continuous functions > > > R^2 --> R, even polynomial ones, which are not surjective.
> > OK. Can you give me an example?
> An example of what? Of a non-surjective function > from R^2 to R?
Gc <Gcut...@hotmail.com> wrote: > On 28 elo, 01:38, Mariano Suárez-Alvarez > <mariano.suarezalva...@gmail.com> wrote: > > On Aug 27, 7:07 pm, Gc <Gcut...@hotmail.com> wrote:
> > > On 27 elo, 23:25, tc...@lsa.umich.edu wrote:
> > > > I recently learned from the book "Mathematical Omnibus" by Fuchs and > > > > Tabachnikov that problem A1 on the 1969 Putnam turned out to be much > > > > harder than intended because the people setting the exam had an > > > > incorrect "solution" in mind. Only 1% of the test takers scored 8/10 > > > > or higher.
> > > > The crux of the matter was this: Let f be a polynomial function from > > > > R^2 to R, i.e., a polynomial in two real variables x and y with real > > > > coefficients. Is it possible for the range of f to be the open interval > > > > (0, infinity)?
> > > > Although there is a short, elementary solution, it is surprisingly > > > > tricky to come up with it.
> > > I`m not an expert, but I try anyway. R^2 is both open and closed, and > > > because a continuous mapping (polynomial) maps open sets to open sets > > > and closed to closed sets, there is no example where a closed set R^2 > > > gets mapped to a non-closed set (0,oo).
> > If your argument were correct, the image of *every* > > continuous real function defined on R^2 would be both open and > > closed and hence, since R is connected, surjective. > > There are, though, examples of continuous functions > > R^2 --> R, even polynomial ones, which are not surjective.
