SageMath MCP Server

polynomial_ring_operation

Idempotent

Construct polynomial rings and compute Groebner bases, ideal dimensions, varieties, reductions, and Groebner checks from variable lists and polynomial strings.

Instructions

Polynomial ring operations: construct rings and compute Groebner bases, ideals, quotients

Input Schema

TableJSON Schema

Name	Required	Description	Default
`session`	No	Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'.	default
`base_ring`	No	Base ring	QQ
`operation`	Yes	One of: groebner_basis, ideal_dimension, ideal_variety, reduce, is_groebner
`ring_vars`	Yes	Variable names, e.g. ['a', 'b', 'c']
`polynomials`	Yes	Polynomials as strings, e.g. ['a^2+b', 'b^2-1']

Output Schema

TableJSON Schema

Name	Required	Description	Default
No arguments

Schema Changelog

Changes observed during successful MCP inspections.

Changed1 schema field changedv0.7.0Sep 12, 2026

changedInput schema / properties / session / description

Previous value: -"Named workspace to use. Workspaces have independent variables; omit for 'default'."New value: +"Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'."

Changed1 schema field changedv0.5.0Aug 14, 2026

addedInput schema / properties / session

Added value: +{
+  "default": "default",
+  "description": "Named workspace to use. Workspaces have independent variables; omit for 'default'.",
+  "type": "string"
+}

First observedv0.3.1Jul 31, 2026

TDQS

C2.9/5.0

Behavior2/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already declare idempotentHint=true, destructiveHint=false, and readOnlyHint=false, so the mutation profile is partly covered. The description adds nothing beyond them – it does not mention that ring construction populates a session workspace with independent variables, nor that results depend on the chosen workspace. For a tool whose state lives in a session, this leaves the key behavioral trait to the schema.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness4/5

Is the description appropriately sized, front-loaded, and free of redundancy?

A single front-loaded sentence with no padding – it leads with the resource and follows with the capability list. It is appropriately sized, though the phrasing 'Polynomial ring operations:' is slightly redundant with the tool name.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness3/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

An output schema exists, so return values need not be explained. However, for a multi-operation tool with five operations, the description never clarifies what reduce, is_groebner, ideal_dimension, or ideal_variety actually compute, leaving the agent to infer semantics from terse names.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters3/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema description coverage is 100%, including a detailed account of the session/workspace parameter, so the schema carries the semantic load. The description's operation list restates values already enumerated in the operation field's documentation and adds no syntax or format detail. Baseline 3 is appropriate.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose4/5

Does the description clearly state what the tool does and how it differs from similar tools?

The description names a specific resource (polynomial rings) and concrete capabilities (construct rings, Groebner bases, ideals, quotients), which distinguishes it from computational siblings like matrix_operation or group_operation. It is clear enough that an agent can identify the domain, though it does not spell out each operation's meaning.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines2/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

There is no guidance on when to choose this over alternatives such as simplify_expression or solve_equation, and no prerequisites or exclusions are stated. The mention of operations describes what can be done, not when to reach for this tool.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.