add buchberger algorithm

src/poly/buchberger.rs

@@ -0,0 +1,74 @@
+use super::flat::Poly;
+use super::var::Var;
+
+/// Computes a Gröbner basis for the ideal generated by `generators` using
+/// Buchberger's algorithm under lex order.
+///
+/// The returned basis spans the same ideal as the input and satisfies
+/// Buchberger's criterion: every S-polynomial of a pair in the basis
+/// reduces to zero modulo the basis.
+pub fn groebner_basis<V: Var>(generators: Vec<Poly<V>>) -> Vec<Poly<V>> {
+    let mut g: Vec<Poly<V>> = generators.into_iter().filter(|p| !p.is_zero()).collect();
+
+    let mut i = 0;
+    while i < g.len() {
+        let mut j = i + 1;
+        while j < g.len() {
+            let s = g[i].s_poly(&g[j]);
+            let r = reduce(&s, &g);
+            if !r.is_zero() {
+                g.push(r);
+            }
+            j += 1;
+        }
+        i += 1;
+    }
+
+    g
+}
+
+/// Checks whether `basis` satisfies Buchberger's criterion under lex order:
+/// the S-polynomial of every pair reduces to zero modulo the basis.
+///
+/// Returns `true` iff `basis` is a Gröbner basis for the ideal it generates.
+pub fn is_groebner_basis<V: Var>(basis: &[Poly<V>]) -> bool {
+    for i in 0..basis.len() {
+        for j in (i + 1)..basis.len() {
+            let s = basis[i].s_poly(&basis[j]);
+            if !reduce(&s, basis).is_zero() {
+                return false;
+            }
+        }
+    }
+    true
+}
+
+/// Reduces `f` modulo `basis` until no leading term of `f` is divisible
+/// by the leading monomial of any element in `basis`.
+///
+/// Uses the pseudo-division remainder and repeats until stable.
+fn reduce<V: Var>(f: &Poly<V>, basis: &[Poly<V>]) -> Poly<V> {
+    let mut p = f.clone();
+
+    'outer: loop {
+        let Some((lm_p, _)) = p.leading_term_lex() else {
+            break;
+        };
+
+        for g in basis {
+            let Some((lm_g, _)) = g.leading_term_lex() else {
+                continue;
+            };
+
+            if lm_p.contains(&lm_g) {
+                let (_, _, r) = p.clone().div_rem(g);
+                p = r;
+                continue 'outer;
+            }
+        }
+
+        break;
+    }
+
+    p
+}