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In this work we compute the size of the fibers of the spectral maps ${\\rm Spec}({\\tt j})_1:{\\rm Spec}({\\mathcal S}(N))\\to{\\rm Spec}({\\mathcal S}(M))$ and ${\\rm Spec}({\\tt j})_2:{\\rm Spec}({\\mathcal S}^*(N))\\to{\\rm Spec}({\\mathcal S}^*(M))$ induced by the inclusion ${\\tt j}:N\\hookrightarrow M$ of a semialgebraic subset $N$ of $M$. 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