20. Influenza_MD2 [129, Figure 3 and Appendix A].

Original generating set information: 7 indeterminates; 28 non-constant functions; maximal total degrees of numerator and denominator are (3, 0); 396 bytes total in string representation.

Indeterminates: c, p, r, c_T, d_I, s_T, β.

Original generating set:

$$c, c, -c, -c, d_I, d_I, -d_I, -d_I, \beta, \beta, \beta c, \beta c, \frac{1000}{11}, \frac{1000}{11}, d_I s_T, c d_I s_T, \frac{-1000}{11} r, \frac{-1000}{11} r,$$

$$\frac{-11}{1000} d_I, \frac{-11}{1000} d_I, \frac{-11}{1000} c d_I, \frac{-11}{1000} c d_I, \beta d_I + d_I r, \beta d_I + d_I r, -c_T d_I + d_I s_T,$$

$$\beta c d_I + c d_I r, -c c_T d_I + c d_I s_T, \frac{1000}{11} c_T - \frac{1000}{11} s_T, \frac{-1000}{11} s_T, \beta c d_I + c d_I r.$$

Result of our algorithm:

$$c, r, c_T, d_I, s_T, \beta.$$

Result is algebraically independent over $\mathbb{C}$: yes.

21. Influenza_MD3 [129, Figure 3 and Appendix A].

Original generating set information: 7 indeterminates; 15 non-constant functions; maximal total degrees of numerator and denominator are (3, 0); 183 bytes total in string representation.

Indeterminates: c, p, r, d_I, d_T, s_T, β.

Original generating set:

$$c, -c, -r, d_I, d_T, -d_I, -s_T, \beta, \beta c, d_I s_T, -d_I d_T, c d_I s_T, -c d_I d_T, \beta d_I + d_I r, \beta c d_I + c d_I r.$$

Result of our algorithm:

$$c, r, d_I, d_T, s_T, \beta.$$

Result is algebraically independent over $\mathbb{C}$: yes.

22. Influenza_MD4 [129, Figure 3 and Appendix A].

Original generating set information: 8 indeterminates; 30 non-constant functions; maximal total degrees of numerator and denominator are (3, 0); 341 bytes total in string representation.

Indeterminates: c, p, r, c_T, d_I, d_T, s_T, β.

Original generating set:

$$c, c, -1, -1, -c, -c, -r, -r, d_I, d_I, d_T, d_T, -d_I, -d_I, -s_T, \beta, \beta, \beta c, \\ \beta c, d_I s_T, -d_I d_T, -d_I d_T, c d_I s_T, c_T - s_T, -c d_I d_T, -c d_I d_T, \beta d_I + d_I r, \\ \beta d_I + d_I r, -c_T d_I + d_I s_T, \beta c d_I + c d_I r, \beta c d_I + c d_I r, -c c_T d_I + c d_I s_T.$$

Result of our algorithm:

$$c, r, c_T, d_I, d_T, s_T, \beta.$$

Result is algebraically independent over $\mathbb{C}$: yes.

23. JAK-STAT [109, Model L1236 in Supplementary materials] with modifications introduced in [111].

Original generating set information: 22 indeterminates; 1300 non-constant functions; maximal total degrees of numerator and denominator are (14, 6); 141.8 KB total in string representation.

Indeterminates: t_1, t_2, t_3, t_4, t_5, t_6, t_7, t_8, t_9, t_{10}, t_{11}, t_{12}, t_{13}, t_{14}, t_{15}, t_{16}, t_{17}, t_{18}, t_{19}, t_{20}, t_{21}, t_{22}.

Original generating set: too large to be listed.

Result of our algorithm:

$$t_1, t_2, t_3, t_4, t_5, t_6, t_7, t_8, t_9, t_{10}, t_{12}, t_{13}, t_{14}, t_{16}, t_{18}, t_{19}, t_{20}, t_{11} t_{21}, t_{15} t_{21}, t_{17} t_{22}.$$

Result is algebraically independent over $\mathbb{C}$: yes.

24. KD1999 [118], [82, Eq. (7.1)].

Original generating set information: 14 indeterminates; 129 non-constant functions; maximal total degrees of numerator and denominator are (9, 3); 3.8 KB total in string representation.

Indeterminates: E, R, V, Δ, T_a, T_h, U, V_h, c_p, k_0, ρ, Ca_0, c_{ph}, ρ_h.

Original generating set: too large to be listed.

Result of our algorithm:

$$V, T_a, T_h, V_h, C_{a0}, \frac{E}{R}, \frac{\Delta}{U}, \frac{c_p \rho}{U}, \frac{c_{ph} \rho_h}{U}.$$

Result is algebraically independent over $\mathbb{C}$: yes.
25. LLW [85, Example 1].

Original generating set information: 4 indeterminates; 9 non-constant functions; maximal total degrees of numerator and denominator are (3, 0); 248 bytes total in string representation.

Indeterminates: $p_1, p_2, p_3, p_4$.

Original generating set:

$$-1, \frac{1}{2}, \frac{-1}{2}, p_2 p_4, -p_1 - p_3, -p_1 - p_3, \frac{1}{2} p_1 + \frac{1}{2} p_3, \frac{3}{2} p_1 + \frac{3}{2} p_3, \frac{-1}{2} p_1^2 p_3 - \frac{1}{2} p_1 p_3^2, \\ \frac{1}{2} p_1 p_2 p_4 + \frac{1}{2} p_2 p_3 p_4, \frac{1}{2} p_1^2 + p_1 p_3 + \frac{1}{2} p_3^2, \frac{-1}{2} p_1^2 - \frac{3}{2} p_1 p_3 - \frac{1}{2} p_3^2.$$

Result of our algorithm:

$$p_1 p_3, p_2 p_4, p_1 + p_3.$$

Result is algebraically independent over $\mathbb{C}$: yes.

26. Lotka-Volterra.

Original generating set information: 4 indeterminates; 4 non-constant functions; maximal total degrees of numerator and denominator are (3, 0); 90 bytes total in string representation.

Indeterminates: $a, b, c, d$.

Original generating set:

$$d, -1, -ab, -ad - bd, a^2 b + ab^2.$$

Result of our algorithm:

$$d, ab, a + b.$$

Result is algebraically independent over $\mathbb{C}$: yes.

27. Lincomp1 [1, Appendix A].

Original generating set information: 7 indeterminates; 6 non-constant functions; maximal total degrees of numerator and denominator are (5, 0); 1.4 KB total in string representation.

Indeterminates: $a_{02}, a_{03}, a_{15}, a_{21}, a_{32}, a_{43}, a_{54}$.

Original generating set: too large to be listed.

Result of our algorithm:

$$\begin{align*} a_{21}a_{32}a_{43}a_{54}, a_{21} + a_{32} + a_{43} + a_{54} + a_{15} + a_{02} + a_{03}, \\ a_{21}a_{32}a_{54}a_{15}a_{03} + a_{21}a_{43}a_{54}a_{15}a_{02} + a_{21}a_{54}a_{15}a_{02}a_{03}, \\ a_{21}a_{32} + a_{21}a_{43} + a_{21}a_{54} + a_{21}a_{15} + a_{21}a_{02} + 14 \text{ more terms}, \\ a_{21}a_{32}a_{43}a_{15} + a_{21}a_{32}a_{54}a_{15} + a_{21}a_{32}a_{54}a_{03} + a_{21}a_{32}a_{15}a_{03} + a_{21}a_{43}a_{54}a_{15} + 10 \text{ more terms}, \\ a_{21}a_{32}a_{43} + a_{21}a_{32}a_{54} + a_{21}a_{32}a_{15} + a_{21}a_{32}a_{03} + a_{21}a_{43}a_{54} + 20 \text{ more terms}. \end{align*}$$

Result is algebraically independent over $\mathbb{C}$: yes.

28. Lincomp2 [1, Appendix A].

Original generating set information: 7 indeterminates; 6 non-constant functions; maximal total degrees of numerator and denominator are (5, 0); 1.4 KB total in string representation.

Indeterminates: $a_{02}, a_{04}, a_{15}, a_{21}, a_{32}, a_{43}, a_{54}$.

Original generating set: too large to be listed.

Result of our algorithm:

$$\begin{align*} a_{21}a_{32}a_{43}a_{54}, a_{21} + a_{32} + a_{43} + a_{54} + a_{15} + a_{02} + a_{04}, \\ a_{21}a_{32}a_{43}a_{15}a_{04} + a_{21}a_{43}a_{54}a_{15}a_{02} + a_{21}a_{43}a_{15}a_{02}a_{04}, \\ a_{21}^2 + a_{32}^2 + 2a_{32}a_{02} + a_{43}^2 + a_{54}^2 + 2a_{54}a_{04} + a_{15}^2 + a_{02}^2 + a_{04}^2, \\ a_{21}a_{32}a_{43}a_{15} + a_{21}a_{32}a_{43}a_{04} + a_{21}a_{32}a_{54}a_{15} + a_{21}a_{32}a_{15}a_{04} + a_{21}a_{43}a_{54}a_{15} + 10 \text{ more terms}, \\ a_{21}a_{32}a_{43} + a_{21}a_{32}a_{54} + a_{21}a_{32}a_{15} + a_{21}a_{32}a_{04} + a_{21}a_{43}a_{54} + 20 \text{ more terms}. \end{align*}$$

Result is algebraically independent over $\mathbb{C}$: yes.

29. Lipolysis [96, Eq. (1)-(5)].

Original generating set information: 3 indeterminates; 12 non-constant functions; maximal total degrees of numerator and denominator are (3, 0); 155 bytes total in string representation.

Indeterminates: $k_2, k_3, k_4$.

Original generating set:

$$k_2, k_2, k_3, -k_2, 2k_4, -k_4^2, k_2 k_3, \frac{1}{3} k_2, 2k_2 k_4, -k_2 k_4^2, \frac{-1}{3} k_3 + k_4, \frac{-1}{3} k_2 k_3 + k_2 k_4.$$
Result of our algorithm:

$k_2, k_3, k_4.$

Result is algebraically independent over $\mathbb{C}$: yes.

30. MAPK-5 [89], [37, Example MAPK pathway and Eq. (6.7) in Appendix].

Original generating set information: 22 indeterminates; 510 non-constant functions; maximal total degrees of numerator and denominator are (8, 6); 633.8 KB total in string representation.

Indeterminates: $a_{00}, a_{01}, a_{10}, b_{00}, b_{01}, b_{10}, c_{0001}, c_{0010}, c_{0011}, c_{0111}, c_{1011}, \beta_{01}, \beta_{10}, \beta_{11}, \alpha_{01}, \alpha_{10}, \alpha_{11}, \gamma_{0100}, \gamma_{1000}, \gamma_{1100}, \gamma_{1101}, \gamma_{1110}$.

Original generating set: too large to be listed.

Result of our algorithm:

$$
\begin{align*}
a_{00}, a_{01}, a_{10}, b_{00}, b_{01}, b_{10}, c_{0001}, c_{0010}, c_{0011}, c_{0111}, c_{1011}, \beta_{01}, \\
\beta_{10}, \beta_{11}, \alpha_{01}, \alpha_{10}, \alpha_{11}, \gamma_{0100}, \gamma_{1000}, \gamma_{1100}, \gamma_{1101}, \gamma_{1110}.
\end{align*}
$$

Result is algebraically independent over $\mathbb{C}$: yes.

31. MAPK-6 [89], [37, Example MAPK pathway and Eq. (6.6) in Appendix].

Original generating set information: 22 indeterminates; 97 non-constant functions; maximal total degrees of numerator and denominator are (5, 3); 17.8 KB total in string representation.

Indeterminates: $a_{00}, a_{01}, a_{10}, b_{00}, b_{01}, b_{10}, c_{0001}, c_{0010}, c_{0011}, c_{0111}, c_{1011}, \beta_{01}, \beta_{10}, \beta_{11}, \alpha_{01}, \alpha_{10}, \alpha_{11}, \gamma_{0100}, \gamma_{1000}, \gamma_{1100}, \gamma_{1101}, \gamma_{1110}$.

Original generating set: too large to be listed.

Result of our algorithm:

$$
\begin{align*}
a_{00}, a_{01}, a_{10}, b_{00}, b_{01}, b_{10}, c_{0001}, c_{0010}, c_{0011}, c_{0111}, c_{1011}, \beta_{01}, \\
\beta_{10}, \beta_{11}, \alpha_{01}, \alpha_{10}, \alpha_{11}, \gamma_{0100}, \gamma_{1000}, \gamma_{1100}, \gamma_{1101}, \gamma_{1110}.
\end{align*}
$$

Result is algebraically independent over $\mathbb{C}$: yes.

32. Ovarian_follicle [6, Section 3.2].

Original generating set information: 22 indeterminates; 1684 non-constant functions; maximal total degrees of numerator and denominator are (21, 10); 44.2 MB total in string representation.

Indeterminates: $n, D_0, D_1, D_2, D_3, D_4, E_0, E_1, E_2, E_3, E_4, J_1, J_2, J_3, f_1, m_1, m_2, m_3, n_1, n_2, n_3, \tau$.

Original generating set: too large to be listed.

Result of our algorithm:

$$
D_2, D_3, D_4, E_0, E_1, E_2, E_3, E_4, J_1, J_2, J_3, m_1, m_2, m_3, n_2, n_3, \tau, n^2, f_1 n, D_0 D_1, D_0 + D_1, f_1 + 2n_1.
$$

Result is algebraically independent over $\mathbb{C}$: yes.

33. Pharm [32], [57, Example 6.4].

Original generating set information: 7 indeterminates; 3533 non-constant functions; maximal total degrees of numerator and denominator are (26, 5); 35.4 MB total in string representation.

Indeterminates: $n, a_1, a_2, b_1, b_2, k_a, k_c$.

Original generating set: too large to be listed.

Result of our algorithm:

$n, a_1, a_2, b_1, b_2, k_a, k_c.$

Result is algebraically independent over $\mathbb{C}$: yes.

34. Pitavastatin [52, Eq. (22)-(24)].

Original generating set information: 8 indeterminates; 249 non-constant functions; maximal total degrees of numerator and denominator are (29, 5); 2.1 MB total in string representation.

Indeterminates: $k, T_0, k_1, k_2, k_3, k_4, r_1, r_3$.

Original generating set: too large to be listed.

Result of our algorithm:

$k_2, k_3, k_4, r_1, r_3, T_0 k, T_0 k_1.$

Result is algebraically independent over $\mathbb{C}$: yes.
35. SEAIJRC [37, Example SEAIJRC in Appendix].

Original generating set information: 7 indeterminates; 1456 non-constant functions; maximal total degrees of numerator and denominator are (31, 8); 26.1 MB total in string representation.

Indeterminates: b, k, q, r, g₁, g₂, α.

Original generating set: too large to be listed.

Result of our algorithm:

$$b, k, g_1, g_2, \alpha, \frac{qr - q}{r}.$$

Result is algebraically independent over $\mathbb{C}$: yes.

36. SEIR1 [131, Eq. (2)].

Original generating set information: 4 indeterminates; 17 non-constant functions; maximal total degrees of numerator and denominator are (7, 1); 1015 bytes total in string representation.

Indeterminates: v, ψ, β, γ.

Original generating set: too large to be listed.

Result of our algorithm:

$$\gamma, \frac{\beta}{\psi}, \psi v - \psi - v, \gamma \psi - \psi v.$$

Result is algebraically independent over $\mathbb{C}$: yes.

37. SEIR2T [113, Example Simple SEIR].

Original generating set information: 4 indeterminates; 37 non-constant functions; maximal total degrees of numerator and denominator are (5, 2); 743 bytes total in string representation.

Indeterminates: N, a, b, v.

Original generating set: too large to be listed.

Result of our algorithm:

$$N, a, b, v.$$

Result is algebraically independent over $\mathbb{C}$: yes.

38. SEIR34 [90, Table 2, ID 34].

Original generating set information: 7 indeterminates; 9 non-constant functions; maximal total degrees of numerator and denominator are (4, 2); 368 bytes total in string representation.

Indeterminates: k, N, r, μ, β, γ, ε.

Original generating set:

$$-1, -N, \frac{\beta r}{kN}, \frac{-\beta \epsilon r}{N}, \epsilon + \gamma + 3\mu, -\epsilon - \gamma - 2\mu, \epsilon \mu + \gamma \mu + 2\mu^2, \epsilon \gamma \mu + \epsilon \mu^2 + \gamma \mu^2 + \mu^3, \\ \frac{\beta \epsilon r + \beta \gamma r + 2\beta \mu r}{kN}, \frac{\beta \epsilon \gamma r + \beta \epsilon \mu r + \beta \gamma \mu r + \beta \mu^2 r}{kN}.$$

Result of our algorithm:

$$N, \mu, K \epsilon, \epsilon \gamma, \beta \epsilon r, \epsilon + \gamma.$$

Result is algebraically independent over $\mathbb{C}$: yes.

39. SEIR36_ref [90, Table 2, ID 51].

Original generating set information: 13 indeterminates; 58 non-constant functions; maximal total degrees of numerator and denominator are (7, 2); 6.9 KB total in string representation.

Indeterminates: N, s, v, φ, s<sub>d</sub>, β, γ, μ<sub>0</sub>, μ<sub>d</sub>, μ<sub>i</sub>, φ<sub>e</sub>, β<sub>d</sub>, γ<sub>d</sub>.

Original generating set: too large to be listed.

Result of our algorithm:

$$N, s, v, \phi, s_d, \beta, \gamma, \mu_0, \mu_d, \mu_i, \phi_e, \beta_d, \gamma_d.$$

Result is algebraically independent over $\mathbb{C}$: yes.
40. SEIRT [116, Eq. (2.1)].

Original generating set information: 4 indeterminates; 6 non-constant functions; maximal total degrees of numerator and denominator are (3, 1); 155 bytes total in string representation.

Indeterminates: $N, \beta, \alpha, \lambda$.

Original generating set:

$$-1, -N, \frac{\beta}{N}, \alpha + \lambda, -\alpha - \lambda, \frac{\alpha\beta\lambda}{N}, \frac{\alpha\beta + \beta\lambda}{N}.$$

Result of our algorithm:

$$N, \beta, \alpha\lambda, \alpha + \lambda.$$

Result is algebraically independent over $\mathbb{C}$: yes.

41. SEUIR [116, Eq. (5.2)].

Original generating set information: 5 indeterminates; 126 non-constant functions; maximal total degrees of numerator and denominator are (11, 2); 5.1 KB total in string representation.

Indeterminates: $N, d, w, z, \beta$.

Original generating set: too large to be listed.

Result of our algorithm:

$$d, z, \frac{Nw}{\beta}.$$

Result is algebraically independent over $\mathbb{C}$: yes.

42. SIR21 [23, Figure 1].

Original generating set information: 6 indeterminates; 7 non-constant functions; maximal total degrees of numerator and denominator are (3, 1); 146 bytes total in string representation.

Indeterminates: $N, q, r, \mu, p, \beta$.

Original generating set:

$$q, q, -N, \frac{N\mu}{\beta}, \frac{-Np}{\beta}, \frac{N\mu p + N\mu q}{\beta}, \frac{-N\mu p - Npr}{\beta}.$$

Result of our algorithm:

$$N, q, r, \mu, p, \beta.$$

Result is algebraically independent over $\mathbb{C}$: yes.

43. SIR24 [80, Eq. (2.1)].

Original generating set information: 5 indeterminates; 68 non-constant functions; maximal total degrees of numerator and denominator are (11, 2); 5.7 KB total in string representation.

Indeterminates: $K, c, \mu, \phi, \gamma$.

Original generating set: too large to be listed.

Result of our algorithm:

$$K, c\phi, \gamma + \mu.$$

Result is algebraically independent over $\mathbb{C}$: yes.

44. SIR6 [132].

Original generating set information: 4 indeterminates; 4 non-constant functions; maximal total degrees of numerator and denominator are (2, 1); 92 bytes total in string representation.

Indeterminates: $k, N, \beta, \gamma$.

Original generating set:

$$-N, \gamma, \frac{kN}{\beta}, \frac{-kN}{\beta}.$$

Result of our algorithm:

$$N, \gamma, \frac{k}{\beta}.$$

Result is algebraically independent over $\mathbb{C}$: yes.
45. SIRC-forced [22, Eq. (7)-(11)].

Original generating set information: 6 indeterminates; 2544 non-constant functions; maximal total degrees of numerator and denominator are (15, 0); 895.4 KB total in string representation.

Indeterminates: $M, g, b_0, b_1, \mu, \nu$.

Original generating set: too large to be listed.

Result of our algorithm:

$g, b_0, \mu, \nu, M^2$.

Result is algebraically independent over $\mathbb{C}$: yes.

46. SIWR-multiout. [86, Eq. (3) with extra output].

Original generating set information: 7 indeterminates; 134 non-constant functions; maximal total degrees of numerator and denominator are (12, 3); 107.5 KB total in string representation.

Indeterminates: $a, k, \beta_i, \beta_w, \mu, \xi, \gamma$.

Original generating set: too large to be listed.

Result of our algorithm:

$a, k, \beta_i, \beta_w, \mu, \xi, \gamma$.

Result is algebraically independent over $\mathbb{C}$: yes.

47. SIWR-orig. [86, Eq. (3)].

Original generating set information: 7 indeterminates; 761 non-constant functions; maximal total degrees of numerator and denominator are (20, 6); 5.7 MB total in string representation.

Indeterminates: $a, k, \beta_i, \beta_w, \mu, \xi, \gamma$.

Original generating set: too large to be listed.

Result of our algorithm:

$a, k, \beta_i, \beta_w, \mu, \xi, \gamma$.

Result is algebraically independent over $\mathbb{C}$: yes.

48. SLIQR [31, Eq. (4)].

Original generating set information: 6 indeterminates; 39 non-constant functions; maximal total degrees of numerator and denominator are (6, 0); 943 bytes total in string representation.

Indeterminates: $\alpha, \beta, \eta, \gamma, \sigma, N$.

Original generating set: too large to be listed.

Result of our algorithm:

$\beta, \sigma, N, \alpha + \gamma, \alpha\eta\gamma, \alpha\gamma + \eta\gamma\sigma - \gamma\sigma$.

Result is algebraically independent over $\mathbb{C}$: yes.

49. Transfection [41], equations copied from [87].

Original generating set information: 5 indeterminates; 25 non-constant functions; maximal total degrees of numerator and denominator are (6, 1); 524 bytes total in string representation.

Indeterminates: $b, d_1, d_2, d_3, k_{TL}$.

Original generating set:

$$ 
\begin{aligned}
& b, b^3, \frac{-1}{3}, \frac{-2}{3}, \frac{-2}{3}b, \frac{-1}{3}b^2, \frac{1}{3}b^3 d_3, \frac{-b d_2}{k_{TL}}, \frac{-1}{3}b^3 d_3, -b + \frac{1}{3}d_3, \frac{-\frac{1}{3}d_2}{k_{TL}}, \frac{-b^2 d_2}{k_{TL}}, \frac{2}{3}b - \frac{1}{3}d_3, b^2 + \frac{1}{3}bd_3, \\
& -b^2 + \frac{1}{3}bd_3, \frac{-b^2 d_1 d_2}{k_{TL}}, \frac{-b^3 d_1 d_2}{k_{TL}}, \frac{-\frac{1}{3}b^3 d_2}{k_{TL}}, \frac{-\frac{1}{3}bd_1 d_2}{k_{TL}}, \frac{1}{3}b^2 - \frac{2}{3}bd_3, \frac{-\frac{1}{3}b^4 d_1 d_2}{k_{TL}}, \frac{-1}{3}b^3 - \frac{1}{3}b^2 d_3, \\
& \frac{-2}{3}b^3 + \frac{1}{3}b^2 d_3, \frac{-b^2 d_2 - bd_1 d_2}{k_{TL}}, \frac{-b^3 d_2 - b^2 d_1 d_2}{k_{TL}}, \frac{-\frac{1}{3}bd_2 - \frac{1}{3}d_1 d_2}{k_{TL}}, \frac{-\frac{1}{3}b^4 d_2 - \frac{1}{3}b^3 d_1 d_2}{k_{TL}}.
\end{aligned}
$$

Result of our algorithm:

$b, d_1, d_3, \frac{d_2}{k_{TL}}$.

Result is algebraically independent over $\mathbb{C}$: yes.
50. SIRT [123, Eq. (2.3)].

Original generating set information: 5 indeterminates; 11 non-constant functions; maximal total degrees of numerator and denominator are (7, 1); 519 bytes total in string representation.

Indeterminates: a, b, d, g, v.

Original generating set:

$$ 
\begin{aligned}
-1, & \frac{b}{g}, dg + v, -dg - v, \frac{2bdg + 2b\nu}{g}, \frac{ab + bg + b\nu}{g}, adg + dg^2 + dg\nu + v^2, -adg - dg^2 - dg\nu - v^2, \\
& \frac{bd^2 g^2 + 2bdg\nu + b\nu^2}{g}, \frac{2abdg + 3ab\nu + 2bdg^2 + 2bdg\nu + 3bg\nu + 2b\nu^2}{g}, \\
& \frac{abd^2 g^2 v + 2abdg\nu^2 + abv^3 + bd^2 g^3 v + 2bdg^2 \nu^2 + bg\nu^3}{g}, \\
& \frac{abd^2 g^2 + 4abdgv + 3abv^2 + bd^2 g^3 + bd^2 g^2 v + 4bdg^2 v + 2bdg\nu^2 + 3bg\nu^2 + b\nu^3}{g}
\end{aligned}
$$

Result of our algorithm:

$$ \frac{b}{g}, dg + v, a + g + v, av + gv. $$

Result is algebraically independent over $\mathbb{C}$: yes.

51. cLV1 [71], equations copied from [8].

Original generating set information: 15 indeterminates; 98 non-constant functions; maximal total degrees of numerator and denominator are (4, 2); 17.9 KB total in string representation.

Indeterminates: $g_1, g_2, g_3, A_{11}, A_{12}, A_{13}, A_{21}, A_{22}, A_{23}, A_{31}, A_{32}, A_{33}, B_{11}, B_{21}, B_{31}$.

Original generating set: too large to be listed.

Result of our algorithm:

$$ g_1, g_2, g_3, A_{11}, A_{12}, A_{21}, A_{22}, A_{31}, A_{32}, B_{11}, B_{21}, B_{31}, \frac{A_{13}}{A_{33}}, \frac{A_{23}}{A_{33}}. $$

Result is algebraically independent over $\mathbb{C}$: yes.

52. genLV [110, Eq. (2.3)].

Original generating set information: 6 indeterminates; 6 non-constant functions; maximal total degrees of numerator and denominator are (3, 1); 335 bytes total in string representation.

Indeterminates: $r_1, r_2, \beta_{11}, \beta_{12}, \beta_{21}, \beta_{22}$.

Original generating set:

$$ 
\begin{aligned}
& \frac{-\beta_{12} - \beta_{22}}{\beta_{12}}, \frac{-\beta_{12}r_2 + 2\beta_{22}r_1}{\beta_{12}}, \frac{\beta_{12}r_1r_2 - \beta_{22}r_1^2}{\beta_{12}}, \frac{-\beta_{11}^2\beta_{22} + \beta_{11}\beta_{12}\beta_{21}}{\beta_{12}}, \\
& \frac{-\beta_{11}\beta_{12} + 2\beta_{11}\beta_{22} - \beta_{12}\beta_{21}}{\beta_{12}}, \frac{\beta_{11}\beta_{12}r_2 - 2\beta_{11}\beta_{22}r_1 + \beta_{12}\beta_{21}r_1}{\beta_{12}}.
\end{aligned}
$$

Result of our algorithm:

$$ r_1, r_2, \beta_{11}, \beta_{21}, \frac{\beta_{12}}{\beta_{22}}. $$

Result is algebraically independent over $\mathbb{C}$: yes.

53. p53 [36] (see also [119, Eq. (1)-(4)]), equations copied from [111].

Original generating set information: 23 indeterminates; 41 non-constant functions; maximal total degrees of numerator and denominator are (9, 4); 611 bytes total in string representation.

Indeterminates: $p_1, p_3, p_4, p_5, p_6, p_7, p_8, p_9, p_{10}, p_{11}, p_{12}, p_{13}, p_{14}, p_{15}, p_{16}, p_{17}, p_{18}, p_{20}, p_{21}, p_{22}, p_{23}, p_{24}, p_{25}$.

Original generating set: too large to be listed.

Result of our algorithm:

$$ p_1, p_3, p_4, p_5, p_6, p_7, p_8, p_9, p_{10}, p_{11}, p_{12}, p_{13}, p_{14}, p_{15}, p_{16}, p_{17}, p_{18}, p_{20}, p_{21}, p_{23}, p_{24}, p_{25}, p_{22}^4. $$

Result is algebraically independent over $\mathbb{C}$: yes.