case n5_m6: n = 5 variables (x, y, u, v, z), degree m = 6, c = 41/16
certificate file certificate_n5_m6.json, sha256 f9d19d2cf0f28b3baa08429b213dea90f13c99939808b83a6123241256322696
  ok    the certificate file names this case, this value of c and this order of the variables
(A) the cube of the seed form f is a sum of squares
  ok    Gram blocks: 4 square matrices of sizes [5, 5, 5, 4]
  ok    sizes as stated in the note
  ok    their 19 index monomials are distinct and of degree 9 in (x, y, z)
  ok    they are the 19 lattice points of half the Newton polytope of f^3 (not needed for (A))
  ok    every Gram block is symmetric
  ok    polynomial identity  f^3 = sum_i m_i^T G_i m_i  over Q (f^3 has 19 monomials)
        block 1 (size 5): smallest pivot ~ 3.111371e-04
        block 2 (size 5): smallest pivot ~ 3.120584e-04
        block 3 (size 5): smallest pivot ~ 3.138235e-04
        block 4 (size 4): smallest pivot ~ 2.199572e-04
  ok    every Gram block is positive definite (LDL^T pivots > 0; leading principal minors > 0)
  ok    the explicit representation f^3 = sum of 19 terms d_j * (form_j)^2, d_j > 0 rational, expands to f^3
  ok    p^3 and q^3 are f^3 in the variables (x, y, z) and (u, v, z)
        smallest pivot over all Gram blocks ~ 2.199572e-04
(B1) the supports: Newton simplex, the set B, the monomials of (p+q)^3
  ok    the 5 listed vertices are monomials of p+q with positive coefficient
  ok    the vertices are affinely independent
  ok    every one of the 7 monomials of p+q has non-negative barycentric coordinates: New(p+q) is this simplex
  ok    B (degree 9 and the 4 weight inequalities) = lattice points of (3/2) New(p+q), from barycentric coordinates
  ok    |B| = 91 (out of 715 monomials of degree 9)
  ok    (p+q)^3 is a form of degree 18 with 82 monomials
  ok    every monomial X^g of (p+q)^3 satisfies w.g >= 0 for all weight vectors w
  ok    every monomial of (p+q)^3 is of the form X^(a+b) with a, b in B (|B+B| = 847)
  ok    every product of three monomials of p+q (82 products; they contain the monomials of (p_t+q_t)^3 for every t) is of the form X^(a+b) with a, b in B
(B2) the functional Lambda and its moment matrix
  ok    the parity vectors of the monomials of (p+q)^3 span the stated space V (dimension 0 over F_2)
  ok    the table has exactly one entry for each of the 91 monomials X^(a+b), a, b in B, with parity in V, and no other entry
  ok    the products of three monomials of p+q have their exponents in the table: Lambda is defined on (p_t+q_t)^3 for every t
  ok    entries between different classes of B modulo (2, V) are zero (631 nonzero entries in all)
  ok    16 classes, of sizes [3, 3, 3, 3, 3, 4, 4, 4, 4, 7, 7, 9, 9, 9, 9, 10]
  ok    the FULL 91 x 91 matrix M is symmetric and positive definite (exact LDL^T, all 91 pivots > 0)
        smallest pivot of the full matrix ~ 3.750967e-07
  ok    every class block is positive definite (LDL^T pivots > 0; leading principal minors > 0)
        smallest pivot over all class blocks ~ 3.750967e-07
(C) the value on (p+q)^3
        Lambda((p+q)^3) = -29836645080831/20480000000000000 ~ -1.456867436e-03
  ok    Lambda((p+q)^3) < 0
  ok    it equals the fraction printed in the note
        for orientation: Lambda(p^3) ~ 5.805353e-03, Lambda(q^3) ~ 5.805353e-03, Lambda(p^2 q) ~ -2.177929e-03, Lambda(p q^2) ~ -2.177929e-03
  ok    Lambda(p^3) + 3 Lambda(p^2 q) + 3 Lambda(p q^2) + Lambda(q^3) is this value; Lambda(p^3), Lambda(q^3) > 0
(D) the cubic phi(t) = Lambda((p_t+q_t)^3)
  ok    p_t + q_t = S - t R with S, R independent of t, and p + q = S - c R
        phi(t) = a3 t^3 + a2 t^2 + a1 t + a0 with
          a3 = -Lambda(R^3)      = -356711369703/5000000000000 ~ -0.0713422739406
          a2 = 3 Lambda(S R^2)   = 669280274427/1000000000000 ~ 0.6692802744270
          a1 = -3 Lambda(S^2 R)  = -10789274712411/5000000000000 ~ -2.1578549424822
          a0 = Lambda(S^3)       = 23337150265011/10000000000000 ~ 2.3337150265011
  ok    10^13 phi(t) = -713422739406 t^3 + 6692802744270 t^2 + (-21578549424822) t + 23337150265011, as printed in the note
  ok    phi(c) = Lambda((p+q)^3)
  ok    the cubic agrees with a direct evaluation of Lambda((p_t+q_t)^3) at t = -5/3, 0, 7/4, 10/3
  ok    a3 < 0 and a2^2 - 3 a1 a3 = -1.390275e-02 < 0: phi' < 0 on the real line, phi is strictly decreasing
  ok    phi(2.5516680) > 0 > phi(2.5516681) and 2.5516681 < c = 41/16: the unique real root lies in (2.5516680, 2.5516681)
        hence phi(t) < 0 for every t larger than the root, in particular for 2.5516681 <= t <= 41/16
(E) the identities used in Proposition 1.5(a)
  ok    M_c' = M_c + (c - c')(xyz)^2 and the identity of Lemma 2.4 for P1 = M_c, P2 = (c - c')(xyz)^2, at c' = 5/2, 1, 0, -3
RESULT for case n5_m6: ALL CHECKS PASSED

case n7_m4: n = 7 variables (x, y, z, x', y', z', w), degree m = 4, c = 13/4
certificate file certificate_n7_m4.json, sha256 e6e416a4e5d6d8e0e6d7ea0537376f2c9becc87461d8085d95a1e42f411fafc3
  ok    the certificate file names this case, this value of c and this order of the variables
(A) the cube of the seed form f is a sum of squares
  ok    Gram blocks: 4 square matrices of sizes [6, 6, 6, 6]
  ok    sizes as stated in the note
  ok    their 24 index monomials are distinct and of degree 6 in (x, y, z, w)
  ok    they are the 24 lattice points of half the Newton polytope of f^3 (not needed for (A))
  ok    every Gram block is symmetric
  ok    polynomial identity  f^3 = sum_i m_i^T G_i m_i  over Q (f^3 has 35 monomials)
        block 1 (size 6): smallest pivot ~ 8.354331e-03
        block 2 (size 6): smallest pivot ~ 2.576601e-02
        block 3 (size 6): smallest pivot ~ 3.250798e-02
        block 4 (size 6): smallest pivot ~ 8.383918e-03
  ok    every Gram block is positive definite (LDL^T pivots > 0; leading principal minors > 0)
  ok    the explicit representation f^3 = sum of 24 terms d_j * (form_j)^2, d_j > 0 rational, expands to f^3
  ok    p^3 and q^3 are f^3 in the variables (x, y, z, w) and (x', y', z', w)
        smallest pivot over all Gram blocks ~ 8.354331e-03
(B1) the supports: Newton simplex, the set B, the monomials of (p+q)^3
  ok    the 7 listed vertices are monomials of p+q with positive coefficient
  ok    the vertices are affinely independent
  ok    every one of the 9 monomials of p+q has non-negative barycentric coordinates: New(p+q) is this simplex
  ok    B (degree 6 and the 6 weight inequalities) = lattice points of (3/2) New(p+q), from barycentric coordinates
  ok    |B| = 99 (out of 924 monomials of degree 6)
  ok    (p+q)^3 is a form of degree 12 with 165 monomials
  ok    every monomial X^g of (p+q)^3 satisfies w.g >= 0 for all weight vectors w
  ok    every monomial of (p+q)^3 is of the form X^(a+b) with a, b in B (|B+B| = 1428)
  ok    every product of three monomials of p+q (165 products; they contain the monomials of (p_t+q_t)^3 for every t) is of the form X^(a+b) with a, b in B
(B2) the functional Lambda and its moment matrix
  ok    the parity vectors of the monomials of (p+q)^3 span the stated space V (dimension 2 over F_2)
  ok    the table has exactly one entry for each of the 168 monomials X^(a+b), a, b in B, with parity in V, and no other entry
  ok    the products of three monomials of p+q have their exponents in the table: Lambda is defined on (p_t+q_t)^3 for every t
  ok    entries between different classes of B modulo (2, V) are zero (825 nonzero entries in all)
  ok    16 classes, of sizes [3, 3, 3, 3, 3, 3, 3, 3, 3, 10, 10, 10, 10, 10, 10, 12]
  ok    the FULL 99 x 99 matrix M is symmetric and positive definite (exact LDL^T, all 99 pivots > 0)
        smallest pivot of the full matrix ~ 1.247331e-05
  ok    every class block is positive definite (LDL^T pivots > 0; leading principal minors > 0)
        smallest pivot over all class blocks ~ 1.247331e-05
(C) the value on (p+q)^3
        Lambda((p+q)^3) = -22652335069827/320000000000000 ~ -7.078854709e-02
  ok    Lambda((p+q)^3) < 0
  ok    it equals the fraction printed in the note
        for orientation: Lambda(p^3) ~ 2.068965e-01, Lambda(q^3) ~ 2.068965e-01, Lambda(p^2 q) ~ -8.076360e-02, Lambda(p q^2) ~ -8.076360e-02
  ok    Lambda(p^3) + 3 Lambda(p^2 q) + 3 Lambda(p q^2) + Lambda(q^3) is this value; Lambda(p^3), Lambda(q^3) > 0
(D) the cubic phi(t) = Lambda((p_t+q_t)^3)
  ok    p_t + q_t = S - t R with S, R independent of t, and p + q = S - c R
        phi(t) = a3 t^3 + a2 t^2 + a1 t + a0 with
          a3 = -Lambda(R^3)      = -285032825877/5000000000000 ~ -0.0570065651754
          a2 = 3 Lambda(S R^2)   = 6422042875791/10000000000000 ~ 0.6422042875791
          a1 = -3 Lambda(S^2 R)  = -28987219462803/10000000000000 ~ -2.8987219462803
          a0 = Lambda(S^3)       = 45237034859253/10000000000000 ~ 4.5237034859253
  ok    10^13 phi(t) = -570065651754 t^3 + 6422042875791 t^2 + (-28987219462803) t + 45237034859253, as printed in the note
  ok    phi(c) = Lambda((p+q)^3)
  ok    the cubic agrees with a direct evaluation of Lambda((p_t+q_t)^3) at t = -5/3, 0, 7/4, 10/3
  ok    a3 < 0 and a2^2 - 3 a1 a3 = -8.331220e-02 < 0: phi' < 0 on the real line, phi is strictly decreasing
  ok    phi(3.1196392) > 0 > phi(3.1196393) and 3.1196393 < c = 13/4: the unique real root lies in (3.1196392, 3.1196393)
        hence phi(t) < 0 for every t larger than the root, in particular for 3.1196393 <= t <= 13/4
(E) the identities used in Proposition 1.5(a)
  ok    Q_c'(x,y,z,w) = (Q_c + e w^4)(x,y,z,sw), s = c'/c, e = s^-4 - 1, and the identity of Lemma 2.4 for P1 = Q_c, P2 = e w^4, at c' = 3, 1, 1/2
RESULT for case n7_m4: ALL CHECKS PASSED

BOTH CASES: ALL CHECKS PASSED
