Creating a polynomial function with no x-intercept

Question

I have an understanding of polynomials and how to create a function based on the leading coefficient, degrees, x-intercepts, etc. My question is how do i go about creating a polynomial function that has no x-intercepts? I am trying to fit a polynomial function to a set of data points(thousands) on a graph. None of the data points are anywhere near the x-axis.

Answer 1

For one, if you examine the asymptotic behavior of a generic polynomial as $x \rightarrow -\infty$ and $x \rightarrow \infty$, you can see that a necessary condition for the polynomial to have no $x$-intercepts is that the degree of the polynomial must be even. (I can also explain why this must be the case with an algebraic rather than an analytic argument if you'd prefer).

Beyond that, you can prevent any even-degree polynomial from having any $x$-intercepts by making the constant term sufficiently large. I.e. by having a sufficiently large $c_0$ when the polynomial is expressed as $f(x) = c_nx^n + c_{n-1}x^{n-1} + \cdots + c_1x + c_0$ where the $c_k$'s are constants. Note that adding or subtracting from $c_0$ shifts the polynomial vertically up and down.

Answer 2

Probably worth pointing out that squares of polynomials are once again polynomials and are non-negative. Adding at least two of these, with no shared $x$-intercepts as you call them, gives a polynomial that is everywhere positive. As near as I can make out, the solution of Hilbert's 17th problem by Emil Artin shows that any positive polynomial in one variable must be the sum of squares of polynomials. I will be doing some checking on that, an algorithm by Delzell in 1984 should help decide the one-variable case. Other references at http://en.wikipedia.org/wiki/Hilbert%27s_seventeenth_problem

EEDDDIIITTTT: Yes, it works when there is only one variable. A non-negative or positive polynomial $f(x)$ in the single variable $x$ can always be written as the sum of squares of at most two polynomials with real coefficients, i.e. either $$ f(x) = \left( p(x) \right)^2 $$ or $$ f(x) = \left( p(x) \right)^2 + \left( q(x) \right)^2 $$

There's an overview of this at: http://junction.stanford.edu/~lall/data/engr210b_0405/sum_of_squares_2004_11_07_01.pdf with enough information to reconstruct the proof on pdf page 12.

As soon as two variables are involved, we need to sum squares of rational functions. Motzkin's 1967 example is $$ x^4 y^2 + x^2 y^4 + 1 - 3 x^2 y^2, $$ which must be the sum of squares of rational functions but is not the sum of squares of polynomials.