Someone over at /pol/ posted this question >>>/pol/77657495

>>8149427
sum of the squares of coordinates.

>>8149429
It's supposed to be R^n --> (0,\infty) but the sum of squares is zero at the origin.

>>8149430
please restate the question. I still dont understand.

>>8149427

(xy-1)^2+x^2

>>8149430
then no.

>>8149435
Is there a polynomial function P:R^n --> (0,\infty) in n variables that is surjective, i.e. onto (0,\infty)?

>>8149441
The image of such a function is closed in \mathbb{R}, so no

>>8149439

To give more detail:

The polynomial is obviously non-negative, and can't be zero because this would require x=0 and xy=1 for some y.

Now let c>0. If c>=1, pick y=0 and x=sqrt(c-1).

If 0<c<1, then pick x=sqrt(c) and y=1/sqrt(c)

In either case, c=(xy-1)^2+x^2

>>8149439
>>8149448


Yes, very good, I salute you! I thought it wasn't possible, and that seemed to be where the other thread was going.

Nice job anon!

(xy-1)^2+x^2 =2 k^2 when x=k and y=1+1/k, and so it's onto.

>>8149451
>Nice job anon!
are you from reddit?

>>8149451
>it's onto

Filthy plebeian.

>>8149439
Good. Now do it in one variable

>>8150547
OP said from R^n to R+.

>>8150589
He didn't restrict it to n>1 tho

>>8150595
for some n
not for all n.
you dumb fucking faggot.

>>8149439
This works, and extends easily to [math]n[/math] variables for [math]n >1 [/math].

>>8150547
Clearly no such thing.

>>8150547
e^x

>>8150649
e^x is not a polynomial

>>8150694
It's a polynomial of infinite order.

>>8150709
e^x is not a polynomial.

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>>8150709
>infinite order

>>8150716
>>8150732

[eqn]e^x = \sum_n \frac{x^n}{n!}[/eqn]

>>8150740
you might want to lurk more.

Also still not a polynomial.

>>8150746
>composed of terms of [math]x^n[/math]
>not a polynomial
u wot m8

>>8150748
What the fuck is wrong with you, leave this guy alone

>>8150748
that's called a power series.
A polynomial specifically has a finite degree.

>>8150748
Polynomials by definition only have definitely many terms.

>>8150756
>>8150762
>>8150780
https://en.wikipedia.org/wiki/Polynomial#Definition
>A polynomial is an expression that can be built from constants and symbols called indeterminates or variables by means of addition, multiplication and exponentiation to a non-negative power.

Nowhere is the degree restricted to be finite.

>>8150800
and a set if a collection of objects.

that's not the definition.

>>8150800
The finitude of the degree is implicit in the word "addition". Addition is formally a binary operation, so building with addition can yield only finite degree.

>>8150807
>and a set if a collection of objects.
That's exactly the definition of set. Where's the problem?

>>8150813
>Addition is formally a binary operation, so building with addition can yield only finite degree.
Nothing prevents us from applying the binary operation of addition an infinite number of times.

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>>8150649
>>8150709
>>8150740
>>8150748
>>8150800
>>8150831
everyone taking the bait of this guy is a retarded newfag.

>>8150800
>Nowhere is it restricted to be infinite
It is if you've taken a basic course in Ring Theory