oxiz_math/polynomial/

gcd_multivariate_advanced.rs

//! Advanced Multivariate Polynomial GCD
#![allow(missing_docs)] // Under development
//!
//! This module implements sophisticated algorithms for computing the greatest
//! common divisor (GCD) of multivariate polynomials:
//! - Subresultant Polynomial Remainder Sequence (PRS)
//! - Modular GCD with Chinese Remainder Theorem (CRT)
//! - Heuristic GCD for sparse polynomials
//! - Content and primitive part computation

#[allow(unused_imports)]
use crate::prelude::*;
use num_bigint::BigInt;
use num_rational::BigRational;
use num_traits::{One, Zero};

/// Monomial representation (variable -> exponent)
pub type Monomial = FxHashMap<usize, usize>;

/// Multivariate polynomial term
#[derive(Debug, Clone)]
pub struct Term {
    pub coefficient: BigRational,
    pub monomial: Monomial,
}

/// Multivariate polynomial
#[derive(Debug, Clone)]
pub struct MultivariatePolynomial {
    pub terms: Vec<Term>,
    pub num_vars: usize,
}

/// Statistics for GCD computation
#[derive(Debug, Clone, Default)]
pub struct GcdStats {
    pub gcd_computations: u64,
    pub subresultant_steps: u64,
    pub modular_reductions: u64,
    pub crt_reconstructions: u64,
    pub heuristic_attempts: u64,
    pub heuristic_successes: u64,
}

/// Configuration for GCD algorithms
#[derive(Debug, Clone)]
pub struct GcdConfig {
    /// Use heuristic GCD for sparse polynomials
    pub use_heuristic: bool,
    /// Use modular GCD algorithm
    pub use_modular: bool,
    /// Threshold for sparse polynomial detection
    pub sparse_threshold: f64,
}

impl Default for GcdConfig {
    fn default() -> Self {
        Self {
            use_heuristic: true,
            use_modular: true,
            sparse_threshold: 0.5,
        }
    }
}

/// Advanced multivariate GCD computer
pub struct MultivariateGcdComputer {
    config: GcdConfig,
    stats: GcdStats,
}

impl MultivariateGcdComputer {
    /// Create a new GCD computer
    pub fn new(config: GcdConfig) -> Self {
        Self {
            config,
            stats: GcdStats::default(),
        }
    }

    /// Compute GCD of two multivariate polynomials
    pub fn gcd(
        &mut self,
        f: &MultivariatePolynomial,
        g: &MultivariatePolynomial,
    ) -> Result<MultivariatePolynomial, String> {
        self.stats.gcd_computations += 1;

        // Handle trivial cases
        if f.is_zero() {
            return Ok(g.clone());
        }
        if g.is_zero() {
            return Ok(f.clone());
        }

        // Check if polynomials are sparse
        let is_sparse = self.is_sparse(f) && self.is_sparse(g);

        // Try heuristic GCD for sparse polynomials
        if self.config.use_heuristic && is_sparse {
            self.stats.heuristic_attempts += 1;
            if let Ok(gcd) = self.heuristic_gcd(f, g) {
                self.stats.heuristic_successes += 1;
                return Ok(gcd);
            }
        }

        // Use modular GCD if enabled
        if self.config.use_modular {
            return self.modular_gcd(f, g);
        }

        // Fall back to subresultant PRS
        self.subresultant_gcd(f, g)
    }

    /// Check if polynomial is sparse
    fn is_sparse(&self, poly: &MultivariatePolynomial) -> bool {
        if poly.terms.is_empty() {
            return true;
        }

        let max_degree = poly.total_degree();
        let max_terms = (max_degree + 1).pow(poly.num_vars as u32);
        let density = poly.terms.len() as f64 / max_terms as f64;

        density < self.config.sparse_threshold
    }

    /// Heuristic GCD using evaluation and interpolation
    fn heuristic_gcd(
        &mut self,
        f: &MultivariatePolynomial,
        g: &MultivariatePolynomial,
    ) -> Result<MultivariatePolynomial, String> {
        // Evaluate at small integer points
        let evaluation_points = vec![0i64, 1, -1, 2, -2];

        let mut gcd_evaluations = Vec::new();

        for &point in &evaluation_points {
            let f_eval = self.evaluate_at_point(f, point);
            let g_eval = self.evaluate_at_point(g, point);

            // Compute univariate GCD
            let gcd_eval = self.univariate_gcd(&f_eval, &g_eval)?;
            gcd_evaluations.push(gcd_eval);
        }

        // Check if all GCD evaluations have the same degree
        let gcd_degree = gcd_evaluations[0].len();
        if gcd_evaluations.iter().all(|g| g.len() == gcd_degree) {
            // Interpolate to recover multivariate GCD
            self.interpolate_gcd(&gcd_evaluations, &evaluation_points)
        } else {
            Err("Heuristic GCD failed: inconsistent degrees".to_string())
        }
    }

    /// Evaluate polynomial at a specific integer point (for main variable)
    fn evaluate_at_point(&self, _poly: &MultivariatePolynomial, _point: i64) -> Vec<BigRational> {
        // Placeholder: evaluate main variable at point
        // Returns univariate polynomial in remaining variables
        vec![BigRational::one()]
    }

    /// Compute GCD of two univariate polynomials
    fn univariate_gcd(
        &self,
        f: &[BigRational],
        g: &[BigRational],
    ) -> Result<Vec<BigRational>, String> {
        let mut a = f.to_vec();
        let mut b = g.to_vec();

        // Euclidean algorithm
        while !Self::is_zero_poly(&b) {
            let r = Self::poly_rem(&a, &b)?;
            a = b;
            b = r;
        }

        Ok(Self::make_monic(a))
    }

    /// Check if polynomial is zero
    fn is_zero_poly(poly: &[BigRational]) -> bool {
        poly.is_empty() || poly.iter().all(|c| c.is_zero())
    }

    /// Polynomial remainder
    fn poly_rem(
        dividend: &[BigRational],
        divisor: &[BigRational],
    ) -> Result<Vec<BigRational>, String> {
        if Self::is_zero_poly(divisor) {
            return Err("Division by zero polynomial".to_string());
        }

        let mut remainder = dividend.to_vec();
        let lc_divisor = &divisor[0];

        while !Self::is_zero_poly(&remainder) && remainder.len() >= divisor.len() {
            let lc_remainder = &remainder[0];
            let quotient_coeff = lc_remainder / lc_divisor;

            // Subtract quotient_coeff * x^degree_diff * divisor
            // When multiplying by x^degree_diff, coefficients align with leading terms
            for i in 0..divisor.len() {
                remainder[i] = &remainder[i] - &quotient_coeff * &divisor[i];
            }

            // Remove leading zeros
            while !remainder.is_empty() && remainder[0].is_zero() {
                remainder.remove(0);
            }
        }

        Ok(remainder)
    }

    /// Make polynomial monic (leading coefficient = 1)
    fn make_monic(mut poly: Vec<BigRational>) -> Vec<BigRational> {
        if poly.is_empty() || poly[0].is_zero() {
            return poly;
        }

        let lc = poly[0].clone();
        for coeff in &mut poly {
            *coeff = &*coeff / &lc;
        }
        poly
    }

    /// Interpolate GCD from evaluations
    fn interpolate_gcd(
        &self,
        _evaluations: &[Vec<BigRational>],
        _points: &[i64],
    ) -> Result<MultivariatePolynomial, String> {
        // Placeholder: Lagrange interpolation
        Ok(MultivariatePolynomial {
            terms: vec![],
            num_vars: 1,
        })
    }

    /// Modular GCD using Chinese Remainder Theorem
    fn modular_gcd(
        &mut self,
        f: &MultivariatePolynomial,
        g: &MultivariatePolynomial,
    ) -> Result<MultivariatePolynomial, String> {
        // Choose primes for modular reduction
        let primes = vec![32003u64, 32009, 32027, 32029, 32051];

        let mut gcd_images = Vec::new();

        for &prime in &primes {
            self.stats.modular_reductions += 1;

            // Reduce polynomials modulo prime
            let f_mod = self.reduce_modulo(f, prime)?;
            let g_mod = self.reduce_modulo(g, prime)?;

            // Compute GCD in finite field
            let gcd_mod = self.gcd_finite_field(&f_mod, &g_mod, prime)?;
            gcd_images.push((gcd_mod, prime));
        }

        // Reconstruct using Chinese Remainder Theorem
        self.stats.crt_reconstructions += 1;
        self.chinese_remainder_reconstruction(&gcd_images)
    }

    /// Reduce polynomial modulo a prime
    fn reduce_modulo(
        &self,
        poly: &MultivariatePolynomial,
        prime: u64,
    ) -> Result<MultivariatePolynomial, String> {
        let mut reduced_terms = Vec::new();

        for term in &poly.terms {
            // Convert rational coefficient to integer modulo prime
            let num = term.coefficient.numer().clone();
            let den = term.coefficient.denom().clone();

            let coeff_mod = self.rational_mod(&num, &den, prime)?;

            if coeff_mod != 0 {
                reduced_terms.push(Term {
                    coefficient: BigRational::from_integer(BigInt::from(coeff_mod)),
                    monomial: term.monomial.clone(),
                });
            }
        }

        Ok(MultivariatePolynomial {
            terms: reduced_terms,
            num_vars: poly.num_vars,
        })
    }

    /// Compute rational number modulo prime (numerator * inverse(denominator))
    fn rational_mod(&self, num: &BigInt, den: &BigInt, prime: u64) -> Result<u64, String> {
        let num_mod = (num.clone() % BigInt::from(prime)).to_u64_digits().1;
        let den_mod = (den.clone() % BigInt::from(prime)).to_u64_digits().1;

        if den_mod.is_empty() || den_mod[0] == 0 {
            return Err("Denominator is zero modulo prime".to_string());
        }

        let den_inv = self.mod_inverse(den_mod[0], prime)?;
        let num_val = if num_mod.is_empty() { 0 } else { num_mod[0] };

        Ok((num_val * den_inv) % prime)
    }

    /// Compute modular inverse using extended Euclidean algorithm
    fn mod_inverse(&self, a: u64, m: u64) -> Result<u64, String> {
        let (g, x, _) = self.extended_gcd(a as i64, m as i64);

        if g != 1 {
            return Err("Modular inverse does not exist".to_string());
        }

        Ok(((x % m as i64 + m as i64) % m as i64) as u64)
    }

    /// Extended Euclidean algorithm
    fn extended_gcd(&self, a: i64, b: i64) -> (i64, i64, i64) {
        if b == 0 {
            return (a, 1, 0);
        }

        let (g, x1, y1) = self.extended_gcd(b, a % b);
        let x = y1;
        let y = x1 - (a / b) * y1;

        (g, x, y)
    }

    /// Compute GCD in finite field
    fn gcd_finite_field(
        &self,
        f: &MultivariatePolynomial,
        _g: &MultivariatePolynomial,
        _prime: u64,
    ) -> Result<MultivariatePolynomial, String> {
        // Placeholder: use Euclidean algorithm in finite field
        Ok(f.clone())
    }

    /// Chinese Remainder Theorem reconstruction
    fn chinese_remainder_reconstruction(
        &self,
        images: &[(MultivariatePolynomial, u64)],
    ) -> Result<MultivariatePolynomial, String> {
        if images.is_empty() {
            return Err("No images for CRT reconstruction".to_string());
        }

        // Placeholder: reconstruct coefficients using CRT
        Ok(images[0].0.clone())
    }

    /// Subresultant polynomial remainder sequence
    fn subresultant_gcd(
        &mut self,
        f: &MultivariatePolynomial,
        g: &MultivariatePolynomial,
    ) -> Result<MultivariatePolynomial, String> {
        // Convert to univariate in main variable
        let main_var = 0; // Choose main variable

        let f_uni = self.to_univariate(f, main_var);
        let g_uni = self.to_univariate(g, main_var);

        // Compute subresultant PRS
        let mut prs = vec![f_uni, g_uni];
        let mut beta = BigRational::from_integer(BigInt::from(-1));
        let mut psi = BigRational::from_integer(BigInt::from(-1));

        loop {
            self.stats.subresultant_steps += 1;

            let i = prs.len() - 2;
            let j = prs.len() - 1;

            if prs[j].is_zero() {
                break;
            }

            // Compute pseudo-remainder
            let remainder = self.pseudo_remainder(&prs[i], &prs[j])?;

            if remainder.is_zero() {
                break;
            }

            // Update subresultant coefficients
            let delta = prs[i].degree(main_var) - prs[j].degree(main_var);
            let next_remainder = self.scale_by_subresultant(&remainder, &beta, &psi, delta);

            prs.push(next_remainder);

            // Update beta and psi for next iteration
            let lc_j = prs[j].leading_coefficient(main_var);
            beta = (-&lc_j).pow(delta as i32);

            if delta > 1 {
                psi = (-&lc_j).pow((delta - 1) as i32) / psi.pow((delta - 2) as i32);
            } else {
                psi = -lc_j;
            }
        }

        // Last non-zero polynomial is the GCD
        let gcd = prs
            .iter()
            .rev()
            .find(|p| !p.is_zero())
            .ok_or("No non-zero polynomial in PRS")?;

        Ok(gcd.clone())
    }

    /// Convert multivariate polynomial to univariate representation
    fn to_univariate(
        &self,
        poly: &MultivariatePolynomial,
        _main_var: usize,
    ) -> MultivariatePolynomial {
        // Placeholder: treat coefficients as polynomials in remaining variables
        poly.clone()
    }

    /// Compute pseudo-remainder
    fn pseudo_remainder(
        &self,
        dividend: &MultivariatePolynomial,
        _divisor: &MultivariatePolynomial,
    ) -> Result<MultivariatePolynomial, String> {
        // Placeholder: pseudo-division algorithm
        Ok(dividend.clone())
    }

    /// Scale remainder by subresultant coefficients
    fn scale_by_subresultant(
        &self,
        remainder: &MultivariatePolynomial,
        _beta: &BigRational,
        _psi: &BigRational,
        _delta: usize,
    ) -> MultivariatePolynomial {
        // Placeholder: apply subresultant scaling
        remainder.clone()
    }

    /// Get statistics
    pub fn stats(&self) -> &GcdStats {
        &self.stats
    }
}

impl MultivariatePolynomial {
    /// Check if polynomial is zero
    pub fn is_zero(&self) -> bool {
        self.terms.is_empty() || self.terms.iter().all(|t| t.coefficient.is_zero())
    }

    /// Get total degree
    pub fn total_degree(&self) -> usize {
        self.terms
            .iter()
            .map(|t| t.monomial.values().sum())
            .max()
            .unwrap_or(0)
    }

    /// Get degree in a specific variable
    pub fn degree(&self, var: usize) -> usize {
        self.terms
            .iter()
            .map(|t| *t.monomial.get(&var).unwrap_or(&0))
            .max()
            .unwrap_or(0)
    }

    /// Get leading coefficient (for specific variable)
    pub fn leading_coefficient(&self, _var: usize) -> BigRational {
        // Placeholder: return leading coefficient
        self.terms
            .first()
            .map(|t| t.coefficient.clone())
            .unwrap_or_else(BigRational::zero)
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_gcd_computer_creation() {
        let config = GcdConfig::default();
        let computer = MultivariateGcdComputer::new(config);
        assert_eq!(computer.stats.gcd_computations, 0);
    }

    #[test]
    fn test_zero_polynomial_gcd() {
        let mut computer = MultivariateGcdComputer::new(GcdConfig::default());

        let f = MultivariatePolynomial {
            terms: vec![],
            num_vars: 1,
        };
        let g = MultivariatePolynomial {
            terms: vec![Term {
                coefficient: BigRational::from_integer(BigInt::from(5)),
                monomial: Monomial::default(),
            }],
            num_vars: 1,
        };

        let result = computer.gcd(&f, &g).expect("test operation should succeed");
        assert_eq!(result.terms.len(), 1);
    }

    #[test]
    fn test_is_sparse() {
        let computer = MultivariateGcdComputer::new(GcdConfig::default());

        // Create a sparse polynomial: 2 terms in 3 variables with degree 2
        // max_terms = (2+1)^3 = 27, density = 2/27 ≈ 0.074 < 0.5
        let mut mono_x2 = Monomial::default();
        mono_x2.insert(0, 2); // x^2

        let mut mono_z2 = Monomial::default();
        mono_z2.insert(2, 2); // z^2

        let sparse = MultivariatePolynomial {
            terms: vec![
                Term {
                    coefficient: BigRational::one(),
                    monomial: mono_x2,
                },
                Term {
                    coefficient: BigRational::one(),
                    monomial: mono_z2,
                },
            ],
            num_vars: 3,
        };

        assert!(computer.is_sparse(&sparse));
    }

    #[test]
    fn test_univariate_gcd_simple() {
        let computer = MultivariateGcdComputer::new(GcdConfig::default());

        // gcd(x^2 - 1, x - 1) = x - 1
        let f = vec![
            BigRational::one(),  // x^2
            BigRational::zero(), // x
            -BigRational::one(), // constant
        ];
        let g = vec![
            BigRational::one(),  // x
            -BigRational::one(), // constant
        ];

        let result = computer
            .univariate_gcd(&f, &g)
            .expect("test operation should succeed");
        assert!(!result.is_empty());
    }

    #[test]
    fn test_poly_rem() {
        let dividend = vec![
            BigRational::one(), // x^2
            BigRational::zero(),
            -BigRational::one(),
        ];
        let divisor = vec![
            BigRational::one(), // x
            -BigRational::one(),
        ];

        let result = MultivariateGcdComputer::poly_rem(&dividend, &divisor)
            .expect("test operation should succeed");
        // Should get zero remainder (x^2 - 1 = (x+1)(x-1))
        assert!(MultivariateGcdComputer::is_zero_poly(&result));
    }

    #[test]
    fn test_make_monic() {
        let poly = vec![
            BigRational::from_integer(BigInt::from(2)),
            BigRational::from_integer(BigInt::from(4)),
        ];

        let monic = MultivariateGcdComputer::make_monic(poly);
        assert_eq!(monic[0], BigRational::one());
        assert_eq!(monic[1], BigRational::from_integer(BigInt::from(2)));
    }

    #[test]
    fn test_mod_inverse() {
        let computer = MultivariateGcdComputer::new(GcdConfig::default());

        let inv = computer
            .mod_inverse(3, 7)
            .expect("test operation should succeed");
        assert_eq!((3 * inv) % 7, 1);
    }

    #[test]
    fn test_extended_gcd() {
        let computer = MultivariateGcdComputer::new(GcdConfig::default());

        let (g, x, y) = computer.extended_gcd(35, 15);
        assert_eq!(g, 5);
        assert_eq!(35 * x + 15 * y, g);
    }

    #[test]
    fn test_rational_mod() {
        let computer = MultivariateGcdComputer::new(GcdConfig::default());

        let num = BigInt::from(7);
        let den = BigInt::from(3);
        let prime = 11;

        let result = computer
            .rational_mod(&num, &den, prime)
            .expect("test operation should succeed");
        // 7/3 mod 11 = 7 * 4 mod 11 = 28 mod 11 = 6
        assert_eq!(result, 6);
    }

    #[test]
    fn test_polynomial_degree() {
        let mut monomial = Monomial::default();
        monomial.insert(0, 2);
        monomial.insert(1, 3);

        let poly = MultivariatePolynomial {
            terms: vec![Term {
                coefficient: BigRational::one(),
                monomial,
            }],
            num_vars: 2,
        };

        assert_eq!(poly.total_degree(), 5);
        assert_eq!(poly.degree(0), 2);
        assert_eq!(poly.degree(1), 3);
    }
}