There are (many) model-theoretic proofs that $DCF_0$, the theory of differentially closed fields of characteristic zero admits quantifier elimination (see "Model Theory" by Marker, Chapter 4 or "A Course in Model Theory" by Ziegler and Tent, Chapter 3), but I can't find any effective algorithm that takes a formula $\phi$ (over the language of differential rings) and constructs an equivalent quantifier-free formula. Is such an algorithm even known? Alternatively, is there a known effective algorithm for deciding $DCF_0$ that possibly does not use full quantifier elimination?
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1Many sources note that Robinson's original proof of model completeness of $\mathrm{DCF}_0$ relied on Seidenberg's elimination algorithm for systems of differential polynomials. I'm not familiar with the details, but this algorithm should directly give an effective algorithm for quantifier elimination relative to $\mathrm{DCF}_0$. A little googling turned up a few more recent papers on improvements to Seidenberg's algorithm. – Alex Kruckman Nov 05 '22 at 19:39
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1Thanks a lot! Did a better search using your pointers: "Generalized Constraint Solving over Differential Algebras" by Thomas Sturm has all the history and references. – toghrul Nov 05 '22 at 23:20
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1Oh, very nice. Maybe you should write an answer to your own question. – Alex Kruckman Nov 06 '22 at 02:48