add S-polynomials

src/poly/tests.rs

@@ -145,6 +145,61 @@ fn test_mono_div() {
     assert_eq!(a.div(&b), Mono { term: vec![] });
 }
 
+#[test]
+fn test_mono_lcm() {
+    // lcm(x², xy) = x²y
+    let a: Mono<StaticVar> = [("x", 2)].into();
+    let b: Mono<StaticVar> = [("x", 1), ("y", 1)].into();
+    assert_eq!(a.lcm(&b), Mono::from([("x", 2), ("y", 1)]));
+
+    // lcm(x, y) = xy  (disjoint)
+    let a: Mono<StaticVar> = [("x", 1)].into();
+    let b: Mono<StaticVar> = [("y", 1)].into();
+    assert_eq!(a.lcm(&b), Mono::from([("x", 1), ("y", 1)]));
+
+    // lcm(x², x²) = x²  (identical)
+    let a: Mono<StaticVar> = [("x", 2)].into();
+    assert_eq!(a.lcm(&a.clone()), a);
+}
+
+#[test]
+fn test_s_poly() {
+    // S(x², xy) = 0: S-poly of two monomials always vanishes
+    let f: Poly<StaticVar> = [(1, [("x", 2)])].into();
+    let g: Poly<StaticVar> = [(1, [("x", 1), ("y", 1)])].into();
+    assert!(f.s_poly(&g).is_zero());
+
+    // f = x² + y, g = xy + z  (lex: LM(f)=x², LM(g)=xy)
+    // L = x²y, t_f = y, t_g = x, d = gcd(1,1) = 1
+    // S = y*(x²+y) - x*(xy+z) = x²y + y² - x²y - xz = y² - xz
+    let f: Poly<StaticVar> = [
+        (1i32, Mono::from([("x", 2u32)])),
+        (1i32, Mono::from([("y", 1u32)])),
+    ].into_iter().collect();
+    let g: Poly<StaticVar> = [
+        (1i32, Mono::from([("x", 1u32), ("y", 1u32)])),
+        (1i32, Mono::from([("z", 1u32)])),
+    ].into_iter().collect();
+    let expected: Poly<StaticVar> = [
+        (1i32, Mono::from([("y", 2u32)])),
+        (-1i32, Mono::from([("x", 1u32), ("z", 1u32)])),
+    ].into_iter().collect();
+    assert_eq!(f.s_poly(&g), expected);
+
+    // f = 2x + y, g = 3x + z  (same LM=x, d=gcd(2,3)=1)
+    // S = (3/1)*f - (2/1)*g = 3(2x+y) - 2(3x+z) = 3y - 2z
+    let f: Poly<StaticVar> = [(2, [("x", 1)]), (1, [("y", 1)])].into();
+    let g: Poly<StaticVar> = [(3, [("x", 1)]), (1, [("z", 1)])].into();
+    let expected: Poly<StaticVar> = [(3, [("y", 1)]), (-2, [("z", 1)])].into();
+    assert_eq!(f.s_poly(&g), expected);
+
+    // f = 4x, g = 6x  (d = gcd(4,6) = 2)
+    // S = (6/2)*x*4x - (4/2)*x*6x = 3*4x - 2*6x = 12x - 12x = 0
+    let f: Poly<StaticVar> = [(4, [("x", 1)])].into();
+    let g: Poly<StaticVar> = [(6, [("x", 1)])].into();
+    assert!(f.s_poly(&g).is_zero());
+}
+
 fn make_const_poly(c: i32) -> Poly<StaticVar> {
     Poly { mono: [(Mono { term: vec![] }, c)].into_iter().collect() }
 }