Abstract:
In terms of differential generators and differential relations for a finitely generated commutative-associative differential $C$-algebra $A$ (with a unit element) there are studied and determined necessary and sufficient conditions in order under any Taylor homomorphism $\widetilde{\psi}_M\colon A\to\mathbb{C}[[z]]$ the transcendence degree of the image $\widetilde{\psi}_M(A)$ over $C$ does not exceed 1 ($\widetilde{\psi}_M (a)\stackrel{{\rm def}}=\sum\limits_{m=0}^{\infty}\psi_M(a^{(m)})\frac{z^m}{m!}$, where $a \in A$, $M \in {\rm Spec}_{\mathbb{C}}A$ is a maximal ideal in $A$, $a^{(m)}$ a result of $m$-fold application of the signature derivation of the element $a$ and $\psi_M$ the canonic epimorphism $A\to A/M$).
Key words:differential algebra, its rank, Taylor homomorphism, analytic spectrum, trajectory germ, orbit closure, affine algebraic curve.
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