Sturm's Theorem: Examples

In a previous entry, I posted the proof of Sturm's Theorem. This is a method for determining the number of real roots in a given interval.

Today, I will show some examples of its use.

Example 1: f(x) = x⁵ - 3x - 1 in the interval [-2, +2]

The Sturm Chain for this polynomial is:

f₀ = x⁵ - 3x - 1

f₁ = 5x⁴ - 3

f₂ = 12x + 5

f₃ = 1

For x=-2, there are 3 sign changes.

For x=-1, there are 2 sign changes

For x=0, there is 1 sign change

For x=1, there is 1 sign change

For x=2, there is 0 sign changes

So, between -2 and -1, there is 3-2=1 real zero

Between -1 and 0, there is 2-1=1 real zero

Between 0 and 1, there are no 1-1=0 real zeros.

Between 1 and 2, there is 1-0=1 real zero.

In summary, between -2 and +2, there are 3-0=3 real zeros.

Example 2: x⁵ -ax -b when a,b are positive and 4⁴a⁵ is greater than 5⁵b⁴

The Sturm Chain for this polynomial is:

f₀ = x⁵ -ax -b

f₁ = 5x⁴ - a

f₂ = 4ax + 5b

f₃ = 4⁴a⁵ - 5⁵b⁴

For this example, let's look at the interval between -∞ and +∞

For -∞, there are 3 sign changes.

For +∞, there are 0 sign changes.

So, all equations of this form have 3-0=3 real roots.

References

• Heinrich Dorrie (Translated by David Antin), 100 Great Problems of Elementary Mathematics (Dover, 1965)