The field of definition for dynamical systems on Pⁿ

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Let Homⁿd be the set of morphisms of degree d from Pⁿ to itself. For f an element of PGLₙ₊₁, let phiᶠ represent the conjugation action f⁻¹ phi f. Let Mⁿd = Homdⁿ/PGLₙ₊₁ be the moduli space of degree d morphisms of Pⁿ. A field of definition for class of morphisms is a field over which at least one morphism in the class is defined. The field of moduli for a class of morphisms is the fixed field of the set of Galois elements fixing that class. Every field of definition contains the field of moduli. In this article, we give a sufficient condition for the field of moduli to be a field of definition for morphisms whose stabilizer group is trivial.