Paper Citation Record · LEDGER

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent

As of 15 August 2026, this Paper Citation Record lists 13 of 13 outbound references and 0 inbound Pith citation observations for arXiv:2606.25121.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2606.25121 v1

Coverage vector

measured 13 of 13 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-06-25T21:23:10.799502Z

measured 13 of 13 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-15T06:32:42.880941+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

13 of 13 outbound references displayed

verified exact1
verified fuzzy0
unresolved11
parse uncertain0
malformed identifier0
metadata mismatch1

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 8114a940-f814-496f-9752-c3c7e6bbb59d · outbound

This paper cites On the complexity of numerical analysis.SIAM J.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent On the complexity of numerical analysis.SIAM J

Reference 1

Resolution: verified exact; arxiv_id, observed 2026-07-04T19:20:07.193062Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 7f4284f1-3f95-43cb-abe5-3fd0742e6ee1 · outbound

This paper cites Algebraic settings for the problem “P̸= NP?”.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Algebraic settings for the problem “P̸= NP?”

Reference 2

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9de7c484-8d41-4eb6-9806-3528c6b68523 · outbound

This paper cites [B¨ ur01] Peter B¨ urgisser.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent [B¨ ur01] Peter B¨ urgisser

Reference 3

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d4f3e1eb-78db-49e1-a463-07a354c3de8d · outbound

This paper cites Completeness classes in algebraic complexity theory.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Completeness classes in algebraic complexity theory

Reference 4

Resolution: metadata mismatch; arxiv_id, observed 2026-07-04T19:20:07.195921Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 4d2a5507-f52b-4745-9cc0-fea696c07326 · outbound

This paper cites Generalized characteristic polynomials.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Generalized characteristic polynomials

Reference 5

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f9864ef5-6205-455d-a26e-fef99309102f · outbound

This paper cites Demystifying the border of depth-3 algebraic circuits.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Demystifying the border of depth-3 algebraic circuits

Reference 6

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 0d1b0524-04bf-4551-87d0-671ef44c56ec · outbound

This paper cites Separated borders: exponential-gap fanin-hierarchy theorem for approx- imative depth-3 circuits.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Separated borders: exponential-gap fanin-hierarchy theorem for approx- imative depth-3 circuits

Reference 7

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 13ead607-e51a-439b-a247-3c261dbaa992 · outbound

This paper cites Quantifier elimination in the theory of an algebraically-closed field.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Quantifier elimination in the theory of an algebraically-closed field

Reference 8

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c26176cf-5929-416a-bd60-a451ea149480 · outbound

This paper cites [Koi97a] Pascal Koiran.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent [Koi97a] Pascal Koiran

Reference 9

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 85fbee1d-53f9-4565-8ca3-3a855016f012 · outbound

This paper cites Randomized and deterministic algorithms for the dimension of algebraic varieties.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Randomized and deterministic algorithms for the dimension of algebraic varieties

Reference 10

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b7c69857-d8d4-42f8-ab43-a8190efdfeab · outbound

This paper cites Circuits versus trees in algebraic complexity.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Circuits versus trees in algebraic complexity

Reference 11

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f5cefaba-2adc-4d78-bd29-a66321da9709 · outbound

This paper cites Valiant’s model: from exponential sums to exponential products.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Valiant’s model: from exponential sums to exponential products

Reference 12

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 426473e1-f5f9-4fbb-82dc-8150f91f561f · outbound

This paper cites an unresolved cited work.

Intractability of Hilbert's Nullstellensatz implies algebraic hardness of permanent Unresolved cited work

Reference 13

Resolution: unresolved; no resolver link, observed 2026-06-25T21:23:10.799502Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.