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Covariant reconstruction of forms from their invariants

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arXiv:2403.17490v2 Announce Type: replace
Abstract: Let $K$ be an algebraically closed field of characteristic $0$ and $f$ be a homogeneous polynomial. We provide an explicit algorithm, which, given the invariants of a generic $f$ under the action of $\mathrm{GL}_n(K)$ returns a polynomial in the orbit of $f$. We derive a specific algorithm for the reconstruction of generic non-hyperelliptic curve of genus 4 from their invariants, as well as a direct reconstruction algorithm for generic non-hyperelliptic curves of genus 3 from their Dixmier-Ohno invariants.

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