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Multiplicative Distance Functions

A positive definite function $\Phi: \mathbb C[x] \rightarrow [0, \infty)$ is a multiplicative distance function if for any monic $f, g \in \mathbb C[x],$ $\Phi(f g) = \Phi(f) \Phi(g)$ and there exists a homogeneity degree $\eta \geq 0$ such that for any $z \in \mathbb C$, $\Phi(z f) = |z|^{\eta} \Phi(f).$

We associate the coefficient vectors of degree $d$ polynomials with the vector space $\mathbb C^{d+1}$ (indexed from $0$ to $d$) and monic degree $d$ polynomials with the vector space $\mathbb C^d$. When we will write $f \in \mathbb C^{d+1}$ (respectively $\mathbb C^d$) we will mean $f$ is a (respectively monic) polynomial of degree $d$ identified with its vector of coefficients. The degree $d$ unit starbody $\mathcal U_d(\Phi) \subset \mathbb C^{d+1}$ consists of (coefficient vectors of) all degree $d$ polynomials $f$ with $\Phi(f) \leq 1$. The starbody of radius $r > 0$ is given by $$r \mathcal U_d(\Phi) = \{ f \in \mathbb C^{d+1} : \Phi(f) \leq r^{\eta} \}.$$ We will write $mu_{d}$ for Lebesgue measure on $\mathbb C^d$ and refer to this as volume in $d$-dimensional complex Euclidean space.

Thus, the volume of $\mathcal U_d(\Phi)$ is the number $\mu_{d+1}(\mathcal U(\Phi)),$ assuming it exists. The distribution function $h_d(\Phi; \xi) = \mu_d \{ f \in \mathbb C^d : \Phi(f) \leq \xi \},$ assuming it exists, is the monic distribution function for $\PhI$. If $r \mathcal U_d(\Phi) \cap \mathbb C^d$ is a Borel set for all $r > 0$ and $h_d(\Phi; \xi)$ is defined and Lebesgue measurable, then we say $\Phi$ is a measurable multiplicative distance function.

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