polytool

Dense univariate polynomial toolkit for combinatorial research. Provides real-rootedness testing by adaptive subresultant integer root counting, Bézout-matrix interlacing checks, exact Hurwitz stability, log-concavity, gamma-positivity, resultants, Ehrhart/h*-vector conversion, recurrence search for polynomial sequences, finite Athanasiadis--Wagner interlacing matrices, and standard sequences — all with exact arithmetic.

Rename note: this crate was renamed from polynomial-tools to polytool before publication. Use polytool in Cargo manifests, Rust imports, CLI documentation, and MCP server configuration.

Features

• Exact real-rootedness testing by adaptive subresultant integer root counting (the engine is the standalone exact-real-roots crate), Bézout-matrix and Sturm-chain interlacing tests, and exact Hurwitz stability.
• Unimodality, log-concavity, ultra-log-concavity, palindromicity and gamma-positivity (with the gamma expansion), Bernstein-basis expansion, resultants and discriminants.
• Ehrhart/h*-vector conversion, h*-inequalities, cyclic and bicyclic sieving checks, Stapledon decomposition.
• Recurrence search for polynomial sequences (with certified complete spaces, coordinate transformations and implicit differential relations), renewal recognition, and a bundled catalog of recurrence-backed OEIS families.
• Planar-network certificates: TNN triangles, strip digraphs and Pólya frequency rows.
• Finite Athanasiadis--Wagner interlacing matrices, standard sequences (Eulerian, Narayana, Chebyshev, Hermite, ...), and enumeration of real-rooted integer polynomials with a prescribed leading prefix.
• A command-line tool, an MCP server and a browser (WebAssembly) wrapper, all over the same exact-arithmetic library.

Installation

From the repository root:

cargo build --release -p polytool

The CLI binary is at target/release/polytool.

Use polytool --version (or polytool -V) for a single deterministic line:

polytool 0.6.0 (git 0123456789ab)

The version comes from the crate manifest and the lowercase 12-hex-digit Git commit is captured at build time. If the source is built without trustworthy Git metadata, the suffix is (git unavailable) rather than a fabricated revision. Reproducible packaging environments may set POLYTOOL_GIT_COMMIT to a full or at least 12-digit hexadecimal revision; an empty or malformed value deliberately selects the unavailable form.

On the generated standalone main branch, the equivalent command is:

cargo build --release

Using as a crate

The library crate is usable today from another Rust project by depending on this directory:

[dependencies]
polytool = { path = "../polytool" }

From GitHub, depend on the repository's standalone main branch:

[dependencies]
polytool = { git = "https://github.com/PerAlexandersson/polytool.git" }

After publishing to crates.io, the dependency would be:

[dependencies]
polytool = "0.6"

The public API is re-exported from polytool, so a downstream crate can write:

use polytool::{check_weak_interlacing, is_real_rooted};

assert!(is_real_rooted(&[1, 11, 11, 1]));
assert_eq!(check_weak_interlacing(&[-1, 1], &[2, -3, 1]), Some(true));

Coefficient vectors are always in ascending degree order. Most convenience functions accept &[i64]; use the *_bigint_coeffs variants when coefficients may exceed i64. Interlacing functions return Option<bool>: None means the directed degree relation is not valid for that test, not that interlacing failed. For the undirected question "do f and g interlace in some order?", use interlaces_either_order, which returns a plain bool.

More library examples (real-rootedness and interlacing, Athanasiadis--Wagner interlacing matrices, planar networks, polynomial arithmetic, recurrence search, enumeration of real-rooted integer polynomials, ...) are in doc/library.md. API documentation is on docs.rs.

CLI quick start

Polynomials are read from standard input, one per line, as comma-separated integer coefficients in ascending degree order (1, 11, 11, 1 is 1 + 11t + 11t^2 + t^3):

echo "1, 11, 11, 1" | polytool real-rooted
polytool interlacing --json < polys.txt
polytool properties < polys.txt
polytool sequence eulerian 5
polytool oeis generate A008292 --rows 8 --format triangle
polytool recurrence < polys.txt
echo "1, 0, -3, 1" | polytool discriminant

Many commands accept --json for machine-readable output, and large integer coefficients are serialized as strings. See doc/cli.md for the input format and the full command reference.

MCP server and web app

• mcp/ contains a local Model Context Protocol server (polytool-mcp) that exposes the exact routines to MCP clients over stdio. Build, install and client configuration are described in mcp/README.md; the remaining notes on the tool set are in doc/mcp.md.
• web/ contains a browser wrapper over the same Rust code (WebAssembly); see doc/web.md for the arbitrary-precision contract and the build.
• crates/exact-real-roots is the standalone exact root-counting crate that polytool depends on; see crates/exact-real-roots/README.md.

Documentation

• doc/cli.md: CLI input format and general commands (real-rootedness, interlacing, properties, Bernstein basis, sequences, family checks, BKW scouting, resultants, enumeration, benchmarks).
  • doc/cli-oeis.md: bundled OEIS catalog, provenance, adding sequences.
  • doc/cli-networks.md: planar networks, TNN triangles, strip digraphs, Pólya frequency rows.
  • doc/cli-recurrence.md: recurrence search, complete spaces and certificates, machine-readable recurrence JSON.
  • doc/cli-recurrence-transforms.md: coordinate transformations, root-domain information, implicit relations.
  • doc/cli-recurrence-representations.md: search for equivalent representations.
  • doc/cli-ehrhart-sieving.md: Ehrhart and h*-vectors, cyclic and bicyclic sieving, Stapledon decomposition.
• doc/library.md: Rust library usage.
• doc/mcp.md and mcp/README.md: MCP server.
• doc/web.md: web wrapper contract and build.
• doc/development.md: tests and checked examples.
• doc/algorithms.md: algorithm notes.
• doc/bezout-interlacing.md and doc/athanasiadis-wagner-lace.md: mathematical notes.
• CHANGELOG.md: release history.

Support

Report issues in the Git repository, or contact Per Alexandersson (@PerAlexandersson, per.w.alexandersson@gmail.com).