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It is well-known that the number of zeros of a polynomial $P_n(z)$ of degree $n$ is precisely $n$ and $P_n(z)$ can be represented in the form $$ P_n(z)=a_n\prod_{i=1}^n(z-z_i), $$ where $a_n$ is the coefficient of $z^n$ and $z_1,z_2,\cdots,z_n$ are the zeros of $P_n(z)$, and we also know the relations between the zeros and the coefficients (Vi\'{e}te theorem).

A fractional polynomial is a function similar to a polynomial but some of the powers of $z$ are not integers, i.e. $$ P(z)=\sum_{i=0}^ka_iz^{\alpha_i}. $$ Do the similar properties hold for fractional polynomials? i.e.,

1) How many zeros are there?

2) Can we obtain the factorization similar to Hadamard's factorization theorem?

3) Can we obtain the relations between the zeros and the coefficients similar to Vi\'{e}te theorem?

We can easy find some special examples as follow

1) Obviously, the number of zeros of $a_\alpha z^\alpha+a_0$ is infinite when $\alpha$ is an irrational number, where $a_\alpha,a_0\in\mathbf{C}$ nonzero.

2) When all the powers of $z$ are rational numbers, i.e. $\alpha_i=p_i/q_i$, we let $q$ be the least common multiple of $q_i$, and let $\lambda=z^\frac{1}{q}$, then it reduces to the case of polynomials.

In fact, I want some concrete examples such as $a_1z^{\alpha+1}+a_2z^\alpha+a_3(\alpha>1)$.

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1 Answer 1

I will try to present some quick facts about exponential polynomials and try to address your questions, however you really ought to look at the literature, particularly starting from Ritt's work in the following articles

[1] J. F. Ritt, "A factorisation theory for functions $\sum_{i=1}^n a_i e^{\alpha_i z}$", Trans. Amer. Math. Soc., 29, 584–596, 1927

[2] J. F. Ritt, "Algebraic combinations of exponentials", Trans. Amer. Math. Soc., 31, 654–679, 1929.

[3] J. F. Ritt, "On the zeros of exponential polynomials", Trans. Amer. Math. Soc., 31, 680–686, 1929

First a few words about factorization of such functions. I will be assuming the underlying field for the coefficients and exponents is $\mathbb C$, though this works over any algebraically closed field. An equivalent representation of these "fractional polynomials" is in the form $\sum_{i} a_i e^{\alpha_i z}$. First note that we have all these units of the form $ae^{\alpha z}$, and moreover, as you noticed, binomials $1-ae^{\alpha z}$ are divisible by $1-a^{1/k}e^{\alpha z/k}$ for all $k\in \mathbb N$, so to have unique factorization one must somehow get around these examples. This was done by Ritt [1] in the case of constant coefficients $a_i$ and by Everst and van der Poorten in "Factorisation in the ring of exponential polynomials", for the general case when $a_i$ are polynomials in $z$.

Let $\mathcal W$ be a finitely generated $\mathbb Z$ submodule of $\mathbb C$, and let $\mathbb C_z\lbrace\mathcal W\rbrace$ denote the ring of exponential polynomials of the form $$E(z)=\sum_{i=0}^m a_i(z)e^{\alpha_i z}$$ where the $\alpha_i$ are distinct. Now one can show that unique factorization holds in this ring.

Theorem: An exponential polynomial $E\in \mathbb C _z\lbrace\mathcal W\rbrace$ factors uniquely up to units as a product of a polynomial $A_0(z)$, a finite number of polynomials $A_i(e^{\beta_i z})$ where the ratio of any two $\beta_i$'s is not rational, and a finite number of exponential polynomials that are irreducible in $\mathbb C _z\lbrace\mathcal W\rbrace$.

In [2] there is a proof that if all zeroes of an exponential polynomial $E$ are also zeroes of the exponential polynomial $F$ then there is an exponential polynomial $G$ and a polynomial $A$ so that $E(z)G(z)=A(z)F(z)$.

Now you should look at [3] for information on zeros of exponential polynomials. Here are two particularly nice facts. The first is an analog of the fact that a polynomial of degree $n$ can have up to $n$ roots. Let $P$ be the smallest convex polygon containing all the frequencies $\alpha_i$ of an exponential polynomial $E$, defined as above. Let the sides of $P$ be denoted as $b_1,b_2,\dots,b_k$.

Theorem: (Tamarkin, Polya and Schwengler) There exist $k$ half strips, with half rays parallel to an outer normal to $b_i$ which together contain all zeroes of $E$. If $|b_i|$ denotes the length of $b_i$, then the number of zeroes in the $i$'th half strip with modulus $\le r$ is asymptotically equal to $\frac{r|b_i|}{2\pi}$.

The second fact (from [3]) is a statement which is an analog to Viete's formula for the case that the product of the roots of a monic polynomial is the constant coefficient, up to sign. Ritt proves that for an exponential polynomial of the form $$f(z)=1+a_1e^{\alpha_1 z}+\cdots +a_ke^{\alpha_k z}$$ with $0 < \alpha_1 < \cdots < \alpha_k$ are real. Let $R(u,v)$ be the sum of the real parts of those roots whose imaginary part lies in $(u,v)$. Then one has $$R(u,v)=-\frac{(v-u)\log|a_k|}{2\pi}+O(1).$$

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Thank you very much for your answer. I am not understand why it equivalents to exponential polynomial. Since $z^\alpha=e^{\alpha\log z}$, it is a little different from $e^{\alpha z}$, I want some concrete examples such as $a_1z^{\alpha+1}+a_2z^\alpha+a_3(\alpha>1)$. –  Yifeng Lu May 2 '12 at 7:23
    
Just let $e^z$ be $z'$ and an exponential polynomial in $z$ becomes a fractional polynomial in $z'$. –  Gjergji Zaimi May 2 '12 at 7:50
    
would rather to see some concrete examples. Would you show the zeros of $z^{\pi+1}+z^\pi+1$ and its factorization –  Yifeng Lu May 2 '12 at 13:27

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