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        "description": "The cited theorem is Section II, §3 in its original homogeneous complex-form setting, with the finite-basis input handling an infinite vanishing family. Homogeneous/projective zero conventions and positive-degree forms matter; it is not stated here as an unrestricted criterion using only real zeros, arbitrary coefficient rings or a uniform exponent independent of the defining system. The exponent depends on the given forms; no efficient modern effective-degree bound is inferred. The preceding normalization and invariant-field arguments are supporting context. The paper's later constructive invariant algorithms and nullform classification lie outside this record.",
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      "statement": "# Hilbert: the homogeneous zero-set criterion behind the Nullstellensatz\n\nHilbert proved a homogeneous polynomial zero-set criterion: if a family of forms vanishes at every common zero of specified forms, then products of sufficiently many members of the vanishing family belong to the ideal generated by the specified forms. In particular, a sufficiently high power of any such form is a polynomial combination of the defining forms. This provides the central bridge from common zeros to ideal powers that became the Nullstellensatz, and supports his analysis of invariant systems.\n\n## Scientific scope\n\nThe cited theorem is Section II, §3 in its original homogeneous complex-form setting, with the finite-basis input handling an infinite vanishing family. Homogeneous/projective zero conventions and positive-degree forms matter; it is not stated here as an unrestricted criterion using only real zeros, arbitrary coefficient rings or a uniform exponent independent of the defining system. The exponent depends on the given forms; no efficient modern effective-degree bound is inferred. The preceding normalization and invariant-field arguments are supporting context. The paper's later constructive invariant algorithms and nullform classification lie outside this record.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nDavid Hilbert\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Ueber die vollen Invariantensysteme](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0042/LOG_0034.pdf)\n",
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      "title": "Hilbert: the homogeneous zero-set criterion behind the Nullstellensatz"
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    "title": "Hilbert: the homogeneous zero-set criterion behind the Nullstellensatz",
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    "what_changed": "Hilbert proved a homogeneous polynomial zero-set criterion: if a family of forms vanishes at every common zero of specified forms, then products of sufficiently many members of the vanishing family belong to the ideal generated by the specified forms. In particular, a sufficiently high power of any such form is a polynomial combination of the defining forms. This provides the central bridge from common zeros to ideal powers that became the Nullstellensatz, and supports his analysis of invariant systems.",
    "scope": "Hilbert proved a homogeneous polynomial zero-set criterion: if a family of forms vanishes at every common zero of specified forms, then products of sufficiently many members of the vanishing family belong to the ideal generated by the specified forms. In particular, a sufficiently high power of any such form is a polynomial combination of the defining forms. This provides the central bridge from common zeros to ideal powers that became the Nullstellensatz, and supports his analysis of invariant systems. The cited theorem is Section II, §3 in its original homogeneous complex-form setting, with the finite-basis input handling an infinite vanishing family. Homogeneous/projective zero conventions and positive-degree forms matter; it is not stated here as an unrestricted criterion using only real zeros, arbitrary coefficient rings or a uniform exponent independent of the defining system. The exponent depends on the given forms; no efficient modern effective-degree bound is inferred. The preceding normalization and invariant-field arguments are supporting context. The paper's later constructive invariant algorithms and nullform classification lie outside this record.",
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