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I have two polynomials

$x = x_1 a + x_3 a^3 + x_5 a^5 ...$

$y = y_1 a + y_3 a^3 + y_5 a^5 ...$

with $a \in \mathbf R(0, \infty)$ and positive real coefficients $x_n$ and $y_n$

$x_n > 0$

$y_n > 0$

Can these two polynomials intersect more than once on the positive half plane? It seems like they intersect at most once for simple equations like $2 x^3$ and $x^5/2$, but does it hold for all positive values of $x_n$ and $y_n$?

Jeremy J
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  • Take $x = a$ and $y = 2 a$. This two polynomials intersect only in the point $a = 0$. – lisyarus Nov 04 '14 at 00:09
  • Subtracting $x$ from $y$, you're essentially asking whether a polynomial (in this case, an odd polynomial) must have a positive root; and that's clearly not required. – Greg Martin Nov 04 '14 at 00:47
  • Greg and lsiyarus you are correct. I rephrased my question. I want to know if they can intersect more than once. – Jeremy J Nov 04 '14 at 01:50

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