High degree polynomial with few real roots

There are univariate polyomials that the number of non-real roots is significantly larger than that of real roots. The simplest example is x^d - 1 where d is odd. It has one real root and d-2 non-real roots. In this post we show an example with exactly three distinct real roots no matter how large is the degree.

Fix three numbers w>u>v>0 and an odd positive integer d \geq 3. Consider the polynomial

f(x) = (w-u)x^d - w^d x + w^d v.

Then this polynomial has exactly three distinct real roots. The author knows two methods to see this. The first one is using elementary calculus. The second method, which is more algebraic, is by applying Sturm’s Theorem.

We illustrate using one example. Let d=7, w = \frac32, u = 1 and v = \frac12. Then f(x) = \frac12 x^7 - \frac{2187}{128}x + \frac{2187}{256}. Next, compute the Sturm sequence:

f_0(x) = f(x) = \frac12 x^7 - \frac{2187}{128}x + \frac{2187}{256},

f_1(x) = f'(x) = \frac72 x^6 - \frac{2187}{128},

f_2(x) = -{\rm remainder}(f_0,f_1) = \frac{6561}{448}x - \frac{2187}{256}, and

f_3(x) = -{\rm remainder}(f_1,f_2) = \frac{101213129}{5971968}.

(These computations can be done easily using the command {\tt quotientRemainder} in {\tt Macaulay 2})

Thus, at x = -\infty, the signs of f_0, \cdots, f_3 are -ve, +ve, -ve, +ve, so there are three sign changes. But there are no sign changes when x=+\infty. Hence the number of distinct real roots is three by Sturm’s Theorem.

The author thinks this kind of phenomenon may be related to the sparsity of the polynomial, which may be something interesting for investigation.

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