/**
 * Exact multivariate polynomial arithmetic + Buchberger's algorithm (B-5).
 *
 * Replaces the former groebnerBasis internals, which (a) extracted coefficients
 * by evaluating at unit points — unable to distinguish `x` from `x²` (both are 1
 * at x = 1) — and (b) never ran Buchberger at all (the "basis" was the parsed
 * inputs). Polynomials are parsed EXACTLY from the expression AST (`parse`), so
 * coefficients are never inferred numerically; division and S-polynomial
 * reduction use graded-free lex order (first variable most significant).
 *
 * Scope: small systems (the CAS surface's documented target). Iteration and
 * basis-size caps throw an honest error instead of returning wrong results.
 */
/** One monomial: coefficient × Π varsᵢ^powersᵢ (dense exponent vector). */
export interface Term {
    coeff: number;
    powers: number[];
}
/** A polynomial: combined, lex-sorted (descending), no near-zero coefficients. */
export type Poly = Term[];
/** Combine like terms, drop |coeff| ≤ EPS, sort lex-descending. */
export declare function normalize(p: Poly): Poly;
/**
 * Return the sum of two multivariate polynomials, with like terms combined.
 * The result drops near-zero terms and is sorted in descending lex order.
 */
export declare function polyAdd(a: Poly, b: Poly): Poly;
/** Return the negation of `a`. The function makes a new polynomial and does not change `a`. */
export declare function polyNeg(a: Poly): Poly;
/** Return the difference `a - b` of two multivariate polynomials. */
export declare function polySub(a: Poly, b: Poly): Poly;
/**
 * Return the product of two multivariate polynomials, with like terms combined.
 * The result drops near-zero terms and is sorted in descending lex order.
 */
export declare function polyMul(a: Poly, b: Poly): Poly;
/**
 * Convert a polynomial expression string to a {@link Poly} over `vars` — exactly.
 * Supports numeric constants, the given variables, `+ − * / ^`, unary minus and
 * parentheses, with `/` restricted to nonzero numeric-constant divisors and `^`
 * to non-negative integer literal exponents. Unknown symbols throw.
 */
export declare function polyFromExpression(expr: string, vars: string[]): Poly;
/**
 * A spatial index grouping divisor polynomials by their leading term's total degree.
 * This filters out polynomials whose degree exceeds the target term, reducing search overhead.
 */
export declare class DivisorGeobucket {
    private buckets;
    private maxDeg;
    constructor(polys: Poly[]);
    insert(poly: Poly): void;
    find(target: number[]): Poly | undefined;
}
/** Remainder of p on multivariate division by G (lex order). */
export declare function polyReduce(p: Poly, G: Poly[] | DivisorGeobucket): Poly;
/**
 * Buchberger's algorithm with honest caps: returns the REDUCED Gröbner basis
 * (monic, minimal, fully inter-reduced) in lex order over the given variable
 * ordering, or throws if the caps are exceeded.
 */
export declare function buchberger(input: Poly[]): Poly[];
/** Format a Poly back to an expression string (constant-first, like `-1 + 1*y + 1*x^2`). */
export declare function polyToString(p: Poly, vars: string[]): string;
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