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POLYNOMIAL HOW TO FIND THE LOWEST AND HIGHEST DEGREE AND THE ORDER

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Ideals and minimal characteristic polynomials

The same sequence can satisfy many different linear recurrences. For example, doubling (5) shows the Fibonacci sequence also satisfies

$\displaystyle 2 F_{n+2} - 2 F_{n+1} - 2F_n = 0,$

which is a linear recurrence with characteristic polynomial

. It also satisfies

$\displaystyle F_{n+3} - F_{n+2} - F_{n+1} = 0,$

and adding these two relations, we find that $\{F_n\}$ also satisfies

$\displaystyle F_{n+3} + F_{n+2} - 3 F_{n+1} - 2 F_n = 0$

which has characteristic polynomial

Now consider an arbitrary sequence $\{a_n\}$ . Let be the set of characteristic polynomials of all linear recurrences satisfied by $\{a_n\}$ . Then (a)

If $f(x) \in I$ and $g(x) \in I$ then $f(x)+g(x) \in I$ .

(b)

If $f(x) \in I$ and

is any polynomial, then $h(x) f(x) \in I$ .

In general, a nonempty set

of polynomials satisfying (a) and (b) is called an ideal.

Fact from algebra: Let be an ideal of polynomials. Then either $I=\{0\}$ or else there is a unique monic polynomial $f(x) \in I$ such that

$\displaystyle I =$ the set of polynomial multiples of $f(x)$ $\displaystyle = \{ h(x)f(x) \mid h(x)$ is a polynomial $\displaystyle \}.$

(A polynomial is monic if the coefficient of the highest power of

is 1.)

This fact, applied to the ideal of characteristic polynomials of a linear recursive sequence $\{a_n\}$ shows that there is always a minimal characteristic polynomial , which is the monic polynomial of lowest degree in . It is the characteristic polynomial of the lowest order non-trivial linear recurrence satisfied by $\{a_n\}$ . The characteristic polynomial of any other linear recurrence satisfied by $\{a_n\}$ is a polynomial multiple of .

The order of a linear recursive sequence $\{a_n\}$ is defined to be the lowest order among all (nontrivial) linear recurrences satisfied by $\{a_n\}$ . The order also equals the degree of the minimal characteristic polynomial. For example, as we showed above, $\{F_n\}$ satisfies

$\displaystyle F_{n+3} + F_{n+2} - 3 F_{n+1} - 2 F_n = 0,$

but we also know that

$\displaystyle F_{n+2}-F_{n+1}-F_n=0,$

and it is easy to show that $\{F_n\}$ cannot satisfy a linear recurrence of order less than 2, so $\{F_n\}$ is a linear recursive sequence of order 2, with minimal characteristic polynomial

Zvezdelina Stankova-Frenkel 2000-09-20