# P2472: Most roots of a degree-30 Littlewood polynomial over F_101 - ID: `P2472` - Reference: `littlewood-f101-max-roots-30` - Page: https://theoremdb.org/statements/P2472 - Export scope: built Markdown snapshot. The current public packet may have changed since this build. - Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e - Current Markdown: https://api.theoremdb.org/v1/statements/littlewood-f101-max-roots-30?representation=markdown - Record maturity: Prospecting problem with recorded work ## The problem For signs \(\varepsilon_1,\ldots,\varepsilon_{30}\in\{-1,1\}\), set \(P_{\varepsilon}(x)=1+\sum_{k=1}^{30}\varepsilon_kx^k\in\mathbb F_{101}[x]\). Determine the largest possible number of distinct roots of \(P_{\varepsilon}\) in \(\mathbb F_{101}\). ### Problem setup - **Definition.** A Littlewood polynomial has every coefficient in {-1,1}. Here the constant coefficient is fixed at 1 by multiplying the whole polynomial by -1 when needed. - **Remark.** Roots are counted without multiplicity. ### What counts as a solution - Give a sign vector attaining the maximum and an exact exhaustive certificate for all 2^30 vectors. ## Status **Resolution status:** Solution submitted. **Lean verification:** Not Lean-verified. A complete enumeration of all 2^30 sign vectors finds two polynomials with 12 roots and none with more. [1](#reference-1) **Recorded scope.** degree-30 Littlewood polynomials over F_101 with constant coefficient 1 [Read this exact record and its supporting evidence](/records/lwf101-claim-exact-maximum-12) ### Verification The lexicographically first maximizing sign vector under the artifact's mask order is \[ (-1,1,1,-1,-1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,1,1,-1,1). \] Its distinct roots in \(\mathbb F_{101}\) are \[ \{20,30,43,47,63,64,66,72,75,93,94,96\}. \] The exhaustive calculation splits the 30 signs into two blocks of 15. It evaluates all 32,768 assignments for each block at all 101 field elements, then checks every one of the \(32{,}768^2=2^{30}\) pairs. The complete root-count histogram has two vectors with 12 roots, 50 with 10 roots, and no vector with 11 roots or more than 12. The exhaustive artifact supplies the full histogram and source. ## Research packet ### Saved solution and evidence **Computation 1 (The exact maximum number of roots is 12).** A complete enumeration of all 2^30 sign vectors finds two polynomials with 12 roots and none with more. The lexicographically first maximizing sign vector under the artifact's mask order is \[ (-1,1,1,-1,-1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,1,1,-1,1). \] Its distinct roots in \(\mathbb F_{101}\) are \[ \{20,30,43,47,63,64,66,72,75,93,94,96\}. \] The exhaustive calculation splits the 30 signs into two blocks of 15. It evaluates all 32,768 assignments for each block at all 101 field elements, then checks every one of the \(32{,}768^2=2^{30}\) pairs. The complete root-count histogram has two vectors with 12 roots, 50 with 10 roots, and no vector with 11 roots or more than 12. The exhaustive artifact supplies the full histogram and source. ### Background and intake notes The certified interval is 9 through 30. Evaluation is cheap for one polynomial, while a complete exclusion still covers more than one billion sign vectors and benefits from parallel blocks. - Original intake status: Status remains unverified. Root distributions of Littlewood polynomials over finite fields are studied, while this exact field and degree may be absent from published tables. - The displayed polynomial gives the lower bound 9, and the degree bound gives an upper bound of 30. - Fixing the constant coefficient leaves 2^30 sign vectors. Bit-sliced evaluation at all 101 field elements can support an exhaustive search, with direct polynomial evaluation retained for every record holder. - Recorded example: The sign vector (-1,1,1,-1,1,-1,-1,-1,-1,1,1,1,1,1,-1,1,-1,1,1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,-1) has roots {28,29,34,49,55,57,60,65,98} modulo 101. ### Prior approaches - **Route 1** (inconclusive): Primary literature treats sparse finite-field polynomials broadly; no table for this Littlewood family was located. [1](#reference-1) ### Runnable artifacts - **Artifact 1** (executable material): The compact C++ program evaluates every sign vector and emits a full histogram whose counts sum to 2^30. [1](#reference-1) ### Computational notes - Exact vectorized evaluation tested 300000 seeded random sign vectors. The best one in that sample had eight roots. - One thousand seeded single-sign-flip annealing starts with 3000 proposals each found the displayed nine-root polynomial. Direct scalar modular evaluation independently reproduced its nine roots. ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `littlewood-f101-max-roots-30`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## Lean verification A solution has been submitted and awaits independent review. 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Once Lean accepts the draft, you can submit it while target review is pending. Your submission request is saved and continues after approval. Private checks stay private until you choose to submit. ## References 1. Zander Kelley, “Roots of Sparse Polynomials over a Finite Field,” LMS Journal of Computation and Mathematics 19 (2016), 196–204. The coefficient-independent bounds for roots of sparse polynomials. https://doi.org/10.1112/S1461157016000334 - Also cited at Complete executable enumeration in lwf101-artifact-exhaustive-mitm - Also cited at Zander Kelley, Roots of Sparse Polynomials over a Finite Field, LMS Journal of Computation and Mathematics 19 (2016), 196-204 - Also cited at Inline C++17 computation executed on 2026-07-24; observed wall time 1.46 seconds on a 10-core arm64 host - journal_article; secondary source; checked 2026-07-28 - Source use: citation_only - Supplies general sparse-polynomial root bounds; the exact degree-30 Littlewood maximum in this packet comes from the exhaustive certificate. - Source named by the research packet.