Matematické Fórum

byk7 · 19. 10. 2016 10:34

Prove or give a counterexample, that every polynomial with positive values (only) in two variables takes its minimum in R^2.

check_drummer · 01. 11. 2016 17:30

Hi, interesting problem. My guess is that the assertion is true. Eg when it can be proved that we can "reach" the point in infinity where the infimum takes place by polynomial substitution x:=f(t),y:=g(t) into the given polynomial P(x,y), then the assesrtion will be proved because we will get polynomial with only one variable (t) and it cannot have a constant limit at infinity.

byk7 · 01. 11. 2016 17:38

:-)

Consider $P(x,y)=x^2+(xy-1)^2$ , we have $x^2\ge0,(xy-1)^2\ge0$ , thus $P(x,y)\ge0$ , but we cannot have $x=0\wedge xy=1$ , so $P(x,y)\gneqq0$ .

Then
$\lim_{t\to\infty}P$\frac{1}{t},t$=0$
so we can get arbitrarily close to zero.

check_drummer

↑ byk7:
I also tried to find something like this - sum of squares - but I failed...

Matematické Fórum

#1 19. 10. 2016 10:34

Positive polynomial

#2 01. 11. 2016 17:30

Re: Positive polynomial

#3 01. 11. 2016 17:38

Re: Positive polynomial

#4 02. 11. 2016 15:25 — Editoval check_drummer (02. 11. 2016 15:26)

Re: Positive polynomial

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