Here’s an elegant little theorem I just learned about. Informally,

A polynomial with few non-zero coefficients has few real roots.

More precisely,

If a polynomial has

knon-zero coefficients, it can have at most 2k– 1 distinct real roots.

This says, for example, that a polynomial like *x*^{100} – 12 *x*^{37} + 1 can have at most 5 distinct real roots, even though it has a total of 100 roots when you count complex roots and count with multiplicity.

I won’t repeat the proof here, but it’s not long or advanced.

The upper bound 2*k* – 1 in the theorem is sharp. To see this, note that

*x*(*x*^{2} – 1)(*x*^{2} – 4) … (*x*^{2} – (*k*-1)^{2})

has *k* non-zero coefficients and 2*k* – 1 distinct real roots.

Source: Mathematical Omnibus

**Update**: Someone asked about the converse. Does having few real roots imply few non-zero coefficients? No. The polynomial (*x*^{2} + 1)* ^{n}* has no real roots but it has

*n*+1 non-zero coefficients.

If I recall correctly this falls right out of some basic calculus. Which is cute. It’s nice to have algebra results informed by calculus.

I feel this is somehow useful in looking at the behaviour of linear recurrences but I don’t remember how.

A nice theorem. Surely a consequence of Descartes’ Rule of Signs

http://www.cut-the-knot.org/fta/ROS2.shtml

Is this true if the coefficients are drawn from C, and not just R?

@Nick so is the question does the upper bound hold for real roots given complex coefficients? The bound clearly doesn’t hold for complex roots.

For real roots I think the result is just a corollary for the theorem with real coefficients. Since a polynomial fin C[X] can just be written as f_1(x) + if_2(x) with f_i in R[X] (where they each have fewer coefficients than f) you must have the same constraint on real roots.