C1  Theorem A bracket (1, sqrt2, sqrt3): fixed polynomials, D = 10; monomials, D = 12
  ok    C1 f = x+y+z  dim ker_p = 11, powers of f = 11: centralizer = polynomials in f (certified)
  ok    C1 f = xy+z^2+x  dim ker_p = 6, powers of f = 6: centralizer = polynomials in f (certified)
  ok    C1 f = x+yz  dim ker_p = 6, powers of f = 6: centralizer = polynomials in f (certified)
  ok    C1 f = x^2y+z^3+y  dim ker_p = 4, powers of f = 4: centralizer = polynomials in f (certified)
  ok    C1 f = xyz+x+y+z  dim ker_p = 4, powers of f = 4: centralizer = polynomials in f (certified)
  ok    C1 f = x^2+y^2+z^2  dim ker_p = 6, powers of f = 6: centralizer = polynomials in f (certified)
  ok    C1 f = xy+yz+zx  dim ker_p = 6, powers of f = 6: centralizer = polynomials in f (certified)
  ok    C1 f = (x+y)^2+z  dim ker_p = 6, powers of f = 6: centralizer = polynomials in f (certified)
  ok    C1 f = x^3+y^3+z^3-3xyz  dim ker_p = 4, powers of f = 4: centralizer = polynomials in f (certified)
  ok    C1 f = x - y  dim ker_p = 11, powers of f = 11: centralizer = polynomials in f (certified)
  ok    C1 f = xy - z^2  dim ker_p = 6, powers of f = 6: centralizer = polynomials in f (certified)
  ok    C1 f = x^2 y^3 z + x  dim ker_p = 2, powers of f = 2: centralizer = polynomials in f (certified)
  ok    C1 f = x (monomial)  dim ker_p = 13 = number of powers of x^(alpha/gcd) of degree <= 12: 13
  ok    C1 f = xy (monomial)  dim ker_p = 7 = number of powers of x^(alpha/gcd) of degree <= 12: 7
  ok    C1 f = x^2 y^3 z (monomial)  dim ker_p = 3 = number of powers of x^(alpha/gcd) of degree <= 12: 3
  ok    C1 f = x^2 y^2 (monomial)  dim ker_p = 7 = number of powers of x^(alpha/gcd) of degree <= 12: 7
  ok    C1 f = z^3 (monomial)  dim ker_p = 13 = number of powers of x^(alpha/gcd) of degree <= 12: 13
C2  random polynomials (2 to 6 terms, degree <= 4, coefficients +-1..3)
  ok    C2 Theorem A bracket, D = 8  500/500 certified: centralizer spanned by the powers of f (of x^gamma0 if f is a polynomial in one monomial x^gamma0)
  ok    C2 Theorem A bracket, D = 10  100/100 certified: centralizer spanned by the powers of f (of x^gamma0 if f is a polynomial in one monomial x^gamma0)
  ok    C2 Example 4.1(b) bracket (no system), D = 8  150/150 certified: centralizer spanned by the powers of f (of x^gamma0 if f is a polynomial in one monomial x^gamma0)
  ok    C2 Example 4.1(d) bracket (no system), D = 8  150/150 certified: centralizer spanned by the powers of f (of x^gamma0 if f is a polynomial in one monomial x^gamma0)
  ok    C2 Theorem D bracket, D = 6  150/150 certified: centralizer spanned by the powers of f (of x^gamma0 if f is a polynomial in one monomial x^gamma0)
  ok    C2 Theorem D bracket, D = 7  30/30 certified: centralizer spanned by the powers of f (of x^gamma0 if f is a polynomial in one monomial x^gamma0)
C3  Theorem D bracket, fixed polynomials, D = 7
  ok    C3 f = x1+x2+x3+x4  dim ker_p = 8, powers of f = 8
  ok    C3 f = x1x2+x3+x4^2  dim ker_p = 4, powers of f = 4
  ok    C3 f = x1x4 - x2x3  dim ker_p = 4, powers of f = 4
  ok    C3 f = x1^2 + x2 x3 x4  dim ker_p = 3, powers of f = 3
  ok    C3 f = x1 x2 x3 x4 + x1  dim ker_p = 2, powers of f = 2
C4  CONTROLS (brackets with polynomial integrable systems: kernels must be larger)
  ok    C4 CONTROL (1,2,3), f = x+y+z  dim ker_p = 16 > 11 powers of f
  ok    C4 CONTROL (1,2,3), f = x  dim ker_p = 21 > 11 powers of f
  ok    C4 CONTROL (1,1,sqrt2), f = y  dim ker_p = 36 > 11 powers of f
  ok    C4 CONTROL (1,1,sqrt2), f = xz + y  dim ker_p = 36 > 6 powers of f
  ok    C4 CONTROL n=4, all lambda_ij = 1, f = x3 + x2x4  dim ker_p = 20 > 4 powers of f
TOTAL failures: 0
running time: 6 s
