\[
\mathrm{Polg}_n(E)(S') = \Bigl\lbrace s_0, s_1, \ldots, s_{n-1}, d_0, d_1,
\ldots, d_{n-1} \Bigm|
\begin{array}{l}
s_i \in E(S') \simeq \Gamma(E_{S'}/S') \\
d_i \in \mathrm{Dr}(E_{S'}) \quad (0 \leqslant i \leqslant n-1)
\end{array} \Bigr.
\]

\[
\Bigl. s_0, s_1 \prec d_0 \,;\ s_1, s_2 \prec d_1 \,;\ \ldots \,;\
s_{n-1}, s_0 \in d_{n-1} \Bigr\rbrace
\]

\[
\mathcal{U}_{\underline{S}_0} = \Bigl\lbrace \bigl( (P_{\underline{s}})_{\underline{s}
\in \underline{S}}, (d_{\underline{a}})_{\underline{a} \in \underline{A}} \bigr)
\in \mathbf{P}\mathrm{ol}_n \Bigm| \ldots \ \forall\, s \in \underline{S}
\Bigr\rbrace
\]

\[
\underbrace{d_0 = d_1 = \cdots = d_i}_{s_0, s_1, \ldots, s_{i+1}} \neq d_{i+1}
\ni s_{i+2}
\]

\[
L_0(\lambda)\,\sigma = 0 \qquad L_1(\lambda)\,\sigma = 0
\qquad\qquad L_0(\lambda) = L_1(\lambda) \qquad L(\lambda)\,\sigma =
\]

\[
u\,\sigma = 0 \qquad\qquad \Sigma \qquad\qquad
\underbrace{L_0 = L_1 = \cdots = L_{n-2}}_{u} \quad L_{n-1}
\]

\[
\underbrace{(L_{n-2} - L_{n-1})}_{M_{n-1}}\, u_{n-1} = u\,.\,\sigma
\]

\[
\begin{aligned}
M_1 u_1 &= (L_0 - L_1)\, u_1 = \lambda_0 u_{-1} u_{12} - \lambda_1 u_1 u_{02}
 + \lambda_2 u_1 u_2 = \\
M_2 u_2 &= (L_1 - L_2)\, u_2 = (\lambda_0 u_{-1} u_{13} - \lambda_1 u_1 u_{03})
 + \lambda_3 u_2 u_3 \quad \mathrm{mod}
\end{aligned}
\]

\[
\lambda_2 u_1 u_2 = -\lambda_0 u_{-1} u_1 + \lambda_1 u_1 (u_0 + u_1)
\]

\[
\underbrace{\lambda_n}_{\lambda_0} u_{n-1} \underbrace{u_n}_{u_0}
= -\lambda_0 \underbrace{u_{-1}}_{u_{n-1}} \underbrace{u_{1n}}_{u_{0n} - u_0}
+ \lambda_1 u_1 u_{0n}
\]

\[
\begin{array}{lll}
u_1 & \lambda_1 u_0 - \lambda_2 u_2 = u & \Rightarrow u_2 \\
u_2 & \lambda_2 u_1 - \lambda_3 u_3 = u & \Rightarrow u_3 \\
u_{n-2} & \lambda_{n-2} u_{n-3} - \lambda_{n-1} u_{n-1} = u & \Rightarrow u_{n-1} \\
u_{n-1} & \lambda_{n-1} u_{n-2} - \lambda_n u_n = u & \Rightarrow u_n = u_0 \\
u_n & \lambda_n u_{n-1} - \lambda_{n+1} u_{n+1} = u & \Rightarrow u_{n+1} = u_1
\end{array}
\]

\[
\mu_0 + \mu_2 = \mu_1 + \mu_3 \qquad\qquad \mu_3 = \mu_0 - \mu_1 + \mu_2
\]

\[
\mathbf{M}_4 \longrightarrow \mathbf{P}^2 \qquad
(u_0, u_1, u_2, u_3) \longmapsto (\underbrace{u_3 \wedge u_0}_{\mu_0},
\underbrace{u_0 \wedge u_1}_{\mu_1}, \underbrace{u_1 \wedge u_2}_{\mu_2})
\]

\[
\begin{aligned}
u_0 \wedge u_2 &= u_0 \wedge (u_0 + u_1) \\
 &= u_0 \wedge (u_1 + u_3) \\
 &= -\underbrace{u_0 \wedge u_1}_{\mu_1} + \underbrace{u_3 \wedge u_0}_{\mu_0}
\end{aligned}
\]

\[
\begin{aligned}
u_0 \wedge u_2 &= \mu_0 - \mu_1 \\
u_1 \wedge u_3 &= \mu_0 - \mu_2 \\
u_2 \wedge u_0 &= \mu_1 - \mu_3 = \mu_1 - \mu_0 = -u_0 \wedge u_2 \\
u_3 \wedge u_1 &= \mu_3 - \mu_0 = \mu_2 - \mu_1 = -u_1 \wedge u_3
\end{aligned}
\]

\[
u_2 = \alpha u_0 + \beta u_1, \qquad u_3 = -(u_0 + u_1 + u_2)
\]

\[
\begin{array}{ll}
u_0 \wedge u_1 = 1\,.\,e & \mu'_1 = 1 \\
u_1 \wedge u_2 = -\alpha\, e & \mu'_2 = -\alpha \\
u_2 \wedge u_3 = (\beta - \alpha)\, e & \mu'_3 = \beta - \alpha \\
u_3 \wedge u_0 = (1 + \beta)\, e & \mu'_0 = 1 + \beta
\end{array}
\qquad
\begin{aligned}
u_2 \wedge u_3 &= -u_2 \wedge u_0 - u_2 \wedge u_1 \\
 &= \beta \underbrace{u_0 \wedge u_1}_{e} - \alpha\, e \\
u_3 \wedge u_0 &= -u_1 \wedge u_0 - u_2 \wedge u_0 \\
 &= e + \beta\, e
\end{aligned}
\]

\[
\alpha = -\frac{\mu_2}{\mu_1} \qquad \beta - \alpha = \frac{\mu_3}{\mu_1}
\qquad \text{i.e.} \qquad \beta = \frac{\mu_3}{\mu_1} + \alpha
= \frac{\mu_3 - \mu_2}{\mu_1}
\]

\[
u_2 = -\frac{\mu_2}{\mu_1} u_0 + \frac{\mu_3 - \mu_2}{\mu_1} u_1
\qquad\qquad \mathbf{M}_4 \simeq \mathbf{P}^2_{\mathbf{Z}}
\]

\[
u_3 = \Bigl( \frac{\mu_2}{\mu_1} - 1 \Bigr) u_0
+ \underbrace{\frac{\mu_2 - \mu_3 - \mu_1}{\mu_1}}_{-\mu_0} u_1
\qquad\qquad
u_2 = 0 \iff \boxed{\mu_2 = 0,\ \mu_3 = 0}\,,\quad \mu_0 = \mu_1
\]

\[
\begin{pmatrix} 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 \end{pmatrix}
\]

\[
\mathcal{F}_\nu \subset T_\nu \subset \widetilde{\Omega}^\nu \qquad
T_\nu = \mathrm{Im}(C \times \Omega^\nu \to \widetilde{\Omega}^\nu)
\]

\[
\mathcal{F}_i \subset T_i \subset \widetilde{\Omega}^i \quad (1 \leqslant i
\leqslant \nu) \qquad\qquad
\mathrm{pr}_{i-1,i}(\mathcal{F}_i) \subset \mathcal{F}_{i-1} \quad
(2 \leqslant i \leqslant \nu)
\]

\[
\mathcal{F}_i = \bigl( (\widetilde{\omega}_1, \ldots, \widetilde{\omega}_i)
\bigm| (\widetilde{\omega}_1, \ldots, \widetilde{\omega}_{i-1}) \in
\mathcal{F}_{i-1},\ \omega_i \ldots
\qquad \widetilde{\Omega}^{i-1} \times \Omega
\]

\[
\mathcal{F}_i = T_i \cap (\widetilde{\Omega}^{i-1} \times \widetilde{\Omega}
\,|\, \mathcal{F}_{i-1} \qquad T_{i-1} \times \Omega
\]

\[
\begin{array}{l}
f_1 \subset \Omega \\
f_2 \subset \Omega \times \Omega \\
\quad \cdots \\
f_i \subset \widetilde{\Omega}^{i-1} \times \Omega, \quad
 f_i \subset T_{i-1} \times \Omega \\
\quad \cdots \\
f_\nu \subset \widetilde{\Omega}^{\nu-1} \times \Omega
\end{array}
\]

\[
\mathcal{F}_i \colon \quad
\mathcal{F}_i = \Bigl\lbrace \bigl( \widetilde{\omega}_1(x), \ldots,
\widetilde{\omega}_{i-1}(x), \omega_i(x) \bigr) \Bigm| x \in C,\
\bigl( \omega_1(x), \ldots, \omega_{i-1}(x), \omega_i \bigr) \in f_i
\Bigr\rbrace
\subset \widetilde{\Omega}^i
\]

\[
\begin{aligned}
f_i &\subset T_{i-1} \times \Omega \subset \widetilde{\Omega}^{i-1}
 \times \Omega \\
\mathcal{F}_i = \widetilde{f}_i &\subset T_i \subset
 \widetilde{\Omega}^{i-1} \times \widetilde{\Omega} = \widetilde{\Omega}^i \\
 &= T_i \cap (\widetilde{\Omega}^i \,|\, f_i)
\end{aligned}
\]

\[
\boxed{\mathcal{F}_1 \subset \Omega} \quad \text{i.e.} \quad
\mathcal{F}_1 \in \mathfrak{P}^*(\Omega)
\]

\[
\widetilde{\mathcal{F}}_1 = (\widetilde{\Omega} \,|\, \mathcal{F}_1) \cap T_1
\subset \widetilde{\Omega} = \widetilde{\Omega}_1
\]

\[
\boxed{f_2 \colon \widetilde{\mathcal{F}}_1 \longrightarrow
\mathfrak{P}^*(\Omega)} \qquad\qquad \widetilde{\Omega} \times \Omega
\]

\[
\underbrace{\widetilde{\mathcal{F}}_2 \subset \widetilde{\Omega}_2 =
\widetilde{\Omega} \times \widetilde{\Omega}} = \bigl\lbrace
(\widetilde{\omega}_1, \widetilde{\omega}_2) \bigm| \omega_1 \in
\mathcal{F}_1,\ (\widetilde{\omega}_1 \times \widetilde{\omega}_2) \in T_2,\
\widetilde{\omega}_2 \in f_2(\widetilde{\omega}_1) \bigr\rbrace
\]

\[
f_3 \colon \widetilde{\mathcal{F}}_2 \longrightarrow \mathfrak{P}(\Omega)
\]

\[
f_\nu \colon \widetilde{\mathcal{F}}_{\nu-1} \longrightarrow
\mathfrak{P}(\Omega) \quad \text{donc} \qquad
\widetilde{\mathcal{F}}_\nu \subset \widetilde{\Omega}^\nu \quad
(\text{en fait, } \widetilde{\mathcal{F}}_\nu \subset T_\nu)
\]

\[
\begin{array}{ll}
\mathcal{F}_1 = f_1 \subset \Omega & \widetilde{\Omega} \times
\widetilde{\Omega} \cdots \times \widetilde{\Omega} \\
f_2 \subset \widetilde{\Omega} \times \Omega & \mathcal{F} \\
f_3 \subset \widetilde{\Omega} \times \widetilde{\Omega} \times \Omega & \\
f_\nu \subset \underbrace{\widetilde{\Omega} \times \widetilde{\Omega}
\cdots \widetilde{\Omega}}_{\nu-1} \times \Omega &
\end{array}
\]

\[
\widetilde{\Omega} = \coprod_{\omega \in \Omega} V_\omega
\xrightarrow{\ \pi\ } \Omega
\]

\[
C(\widetilde{\omega}, x) = f_\omega^{-1}\bigl(\lbrace f_\omega(x)
\rbrace\bigr) = C(\omega(x))
\]

\[
C(\omega_1, \ldots, \omega_\nu; x) = \bigcap_{1 \leqslant i \leqslant \nu}
C(\omega_i; x)
\]

\[
\widetilde{\Omega} \longrightarrow \mathfrak{P}(\mathcal{C}), \quad
\widetilde{\omega} \mapsto C(\widetilde{\omega}) \qquad\qquad
\widetilde{\omega} = (\omega, v) \longmapsto f_\omega^{-1}(\lbrace v \rbrace)
\]

\[
\widetilde{\Omega}^\nu \longrightarrow \mathfrak{P}(C), \qquad
(\widetilde{\omega}_1, \ldots, \widetilde{\omega}_\nu) \longmapsto
C(\widetilde{\omega}_1, \ldots, \widetilde{\omega}_\nu) =
\bigcap_{1 \leqslant i \leqslant \nu} C(\widetilde{\omega}_i)
\]

\[
C \times \Omega \longrightarrow \widetilde{\Omega}, \quad (x, \omega)
\longmapsto \omega(x) \qquad \Longrightarrow \qquad
C \longrightarrow \Gamma(\widetilde{\Omega}/\Omega)
\]

\[
C \times \Omega^\nu \longrightarrow \widetilde{\Omega}^\nu
\]

\[
C \longrightarrow \Gamma(\widetilde{\Omega}^\nu/\Omega^\nu), \qquad
x \longmapsto \bigl( (\omega_1, \ldots, \omega_\nu) \longmapsto
(\omega_1(x), \ldots, \omega_\nu(x)) \bigr)
\]

\[
\left\lbrace
\begin{aligned}
& u = \lambda_i u_{i-1} - \lambda_{i+1} u_{i+1} \qquad i = 0, 1, \ldots, n-1 \\
& \textstyle\sum u_i = 0
\end{aligned}
\right.
\]

\[
\left\lbrace
\begin{aligned}
& u_{i-1} \wedge u_i = \mu_i \\
& u \wedge u_i = \alpha_i
\end{aligned}
\right.
\]

\[
\left\lbrace
\begin{aligned}
& \lambda_i \mu_i + \lambda_{i+1} \mu_{i+1} = \alpha_i \\
& \lambda_i \alpha_{i-1} - \lambda_{i+1} \alpha_{i+1} = 0 \\
& \textstyle\sum \alpha_i = 0
\end{aligned}
\right.
\qquad\qquad 2 \sum \lambda_i \mu_i = 0
\]

\[
u = \frac{\alpha_1}{\mu_1} u_0 - \frac{\alpha_0}{\mu_1} u_1
= \Bigl( \lambda_1 + \lambda_2 \frac{\mu_2}{\mu_1} \Bigr) u_0
- \Bigl( \lambda_0 \frac{\mu_0}{\mu_1} + \lambda_1 \Bigr) u_1
\]

\[
CI_n^* \longrightarrow \mathbf{P}^{n-1} \times \mathbf{P}^{n-1}, \qquad
(\lambda_i) \quad (\mu_i)
\]

\[
u_2 = -\frac{\mu_2}{\mu_1} u_0 + \frac{1}{\lambda_2}
\Bigl( \lambda_0 \frac{\mu_0}{\mu_1} + \lambda_1 \Bigr) u_1
\]

\[
u_{n-1} = A_{n-1}\bigl((\lambda)(\mu)\bigr)\, u_0 + B_{n-1}(\ ,\ )\, u_1
\]

\[
f_i = x_{i-1} y_i - y_{i-1} x_i
\]

\[
df_i = y_i\, dx_{i-1} - x_i\, dy_{i-1} - (y_{i-1}\, dx_i - x_{i-1}\, dy_i)
\]

\[
\sum \lambda_i\, df_i = \sum_i (-\lambda_i y_{i-1} + \lambda_{i+1} y_{i+1})\,
dx_i + \sum_i (\lambda_i x_{i-1} - \lambda_{i+1} x_{i+1})\, dy_i
= \mu \sum dx_i + \nu \sum dy_i
\]

\[
\left.
\begin{aligned}
-\mu &= +\lambda_i y_{i-1} - \lambda_{i+1} y_{i+1} \\
\nu &= +\lambda_i x_{i-1} - \lambda_{i+1} x_{i+1}
\end{aligned}
\right\rbrace \quad 0 \leqslant i \leqslant n-1
\]

\[
\lambda_0, \lambda_1 \ \big|\ \overset{i=1}{\lambda_2} \cdots
\overset{i=n-2}{\lambda_{n-1}} \qquad\qquad
\overset{i=n-1}{\lambda_n} \ \text{relation}
\]

\[
\begin{aligned}
&+\lambda_0 y_{n-1} - \lambda_1 y_1 = +\lambda_1 y_0 - \lambda_2 y_2
= +\lambda_2 y_1 - \lambda_3 y_3 = \cdots
= \lambda_{n-2} y_{n-3} - \lambda_{n-1} y_{n-1} \\
&\qquad = \lambda_3 y_2 - \lambda_4 y_4 \qquad\qquad
= \lambda_{n-1} y_{n-2} - \lambda_n y_n \quad
(\lambda_n = \lambda_0,\ y_n = y_0)
\end{aligned}
\]

\[
\begin{aligned}
(y_2)\,\lambda_2 &= \lambda_1 (y_0 + y_1) - \lambda_0 y_{n-1}
 = \lambda_0 (-y_{n-1}) + \lambda_1 (y_0 + y_1) \qquad
 (= \lambda_4 y_3 - \lambda_5 y_5) \\
(y_2 y_3)\,\lambda_3 &= \underbrace{(y_2 \lambda_2)}\, y_1
 - y_2 (\lambda_0 y_{n-1} - \lambda_1 y_1) \\
 &= \lambda_0 \bigl( -y_{n-1} (y_1 + y_2) \bigr)
 + \lambda_1 y_1 [y_0 + y_1 + y_2] \\
(y_2 y_3 y_4)\,\lambda_4 &= (y_2 y_3 \lambda_3)\, y_3
 - y_2 y_3 (\lambda_0 y_{n-1} - \lambda_1 y_1) \\
 &= \lambda_0 \bigl( -y_2 y_{n-1} (y_1 + y_2 + y_3) \bigr)
 + \lambda_1 y_1 y_2 (y_0 + y_1 + y_2 + y_3) \\
(y_2 y_3 y_4 y_5)\,\lambda_5 &= \underbrace{(y_2 y_3 y_4 \lambda_4)}\, y_3
 - y_2 y_3 y_4 (\lambda_0 y_{n-1} - \lambda_1 y_1)
\end{aligned}
\]

\[
= \lambda_0 \bigl( -y_{n-1} y_2 y_3 (y_1 + y_2 + y_3 + y_4) \bigr)
+ \lambda_1 y_1 y_2 y_3 (y_0 + y_1 + y_2 + y_3 + y_4)
\]

\[
\boxed{
\begin{aligned}
(y_{i-1} y_i)\,\lambda_i &= (-\lambda_0 y_{-1})(y_1 + \cdots + y_{i-1})
 + (\lambda_1 y_1)(y_0 + \cdots + y_{i-1}) \\
(x_{i-1} x_i)\,\lambda_i &= (-\lambda_0 x_{-1})(x_1 + \cdots + x_{i-1})
 + (\lambda_1 x_1)(x_0 + \cdots + x_{i-1})
\end{aligned}}
\]

\[
\boxed{\lambda_i u_{i-1} u_i = -\lambda_0 u_{-1} u_{1i} + \lambda_1 u_1 u_{0i}}
\]

\[
\boxed{
\begin{aligned}
&\lambda_0 \bigl[ -x_{i-1} x_i y_{-1} (y_1 + \cdots + y_{i-1})
 + y_{i-1} y_i x_{-1} (x_1 + \cdots + x_{i-1}) \bigr] \\
&\quad + \lambda_1 \bigl[ x_{i-1} x_i y_1 (y_0 + \cdots + y_{i-1})
 - y_{i-1} y_i x_1 (x_0 + \cdots + x_{i-1}) \bigr] = 0
\end{aligned}}
\qquad i = 2, \ldots, n-1
\]

\[
(\underbrace{y_{n-1}}_{y_{-1}} \underbrace{y_n}_{y_0})
\underbrace{\lambda_n}_{\lambda_0}
= (-\lambda_0 y_{-1}) \underbrace{(y_1 + \cdots + y_{n-1})}_{-y_0}
+ (\lambda_1 y_1) \underbrace{(y_0 + \cdots + y_{n-1})}_{0}
\]

\[
\lambda_0 (y_{-1} y_0) = \lambda_0 y_0 y_{-1} \qquad \text{ok}
\]

\[
\begin{aligned}
&\lambda_0 \bigl( -x_1 x_2 y_{-1} \underbrace{(y_1)}_{-y_0}
 + y_1 y_2 x_{-1} (x_1) \bigr) \\
&\quad + x_1 x_2 y_2 (y_0) - y_1 y_2 x_2 (\ldots) \\
&\quad x_2 y_2 (x_1 y_0 - y_1 x_0)
\end{aligned}
\]

\[
-x_1 x_2 y_1 y_3 + y_1 y_2 x_1 x_3 = -x_1 y_1 (x_2 y_3 - y_2 x_3)
\]

\[
\begin{aligned}
x_1 x_2 y_1 \underbrace{(y_0 + y_1)}_{-y_2 - y_3}
 - y_1 y_2 x_1 \underbrace{(x_0 + x_1)}_{-x_2 - x_3} &= \\
= -x_1 x_2 y_1 (y_2 + y_3) + y_1 y_2 x_1 (x_2 + x_3) & \\
= -x_1 x_2 y_1 y_3 + y_1 y_2 x_1 x_3 &= -x_1 y_1 (x_2 y_3 - y_2 x_3)
\end{aligned}
\qquad\qquad \boxed{\lambda_0 + \lambda_1 = 0}
\]

\[
-x_2 x_3 y_3 (y_1 + y_2) + y_2 y_3 x_3 (x_1 + x_2)
= -x_3 y_3 (x_1 y_2 - y_1 x_2)
\]

\[
x_2 x_3 y_1 \underbrace{(y_0 + y_1 + y_2)}_{-y_3}
- y_2 y_3 x_1 \underbrace{(x_0 + x_1 + x_2)}_{-x_3}
= x_3 y_3 (x_1 y_2 - y_1 x_2)
\qquad\qquad \boxed{\lambda_0 + \lambda_1 = 0}
\]

\[
u_0 \wedge u_1 \quad u_1 \wedge u_2 \quad u_2 \wedge u_3 \qquad
u_3 \wedge u_0 = -(u_0 + u_1 + u_2) \wedge u_0 = u_0 \wedge u_1 +
u_0 \wedge u_2
\]

\[
\begin{array}{ll}
(a - 1, 0) & \bigl( -(a-1), b-1 \bigr) = (-a+1, b-1) \quad u_0 \\
(0, b - 1) & \bigl( a+1, -(b-1) \bigr) = (a+1, -b+1) \quad u_1 \\
(a + 1, 0) & \bigl( -(a+1), b+1 \bigr) = (-a-1, b+1) \quad u_2 \\
(0, b + 1) & \bigl( a-1, -(b+1) \bigr) = (a-1, -b-1) \quad u_3
\end{array}
\]

\[
\begin{aligned}
(1-a)(1-b) + (1-b)(a+1) &= 2(1-b) = f_1 \\
(a+1)(b+1) + (a+1)(1-b) &= 2(a+1) = f_2 \\
(a+1)(b+1) + (b+1)(1-a) &= 2(b+1) = f_3 \\
(a-1)(b-1) - (b+1)(a-1) &= 2(1-a) = f_0
\end{aligned}
\]

\[
f_1 + f_3 = f_2 + f_0 = v \quad \Big/ \quad
\begin{aligned}
f_3 - f_1 &= (b)\, v \\
f_2 - f_0 &= (a)\, v
\end{aligned}
\]

\[
u_0 \wedge u_1 + u_2 \wedge u_3 = u_1 \wedge u_2 + u_3 \wedge u_0
\overset{?}{=} (u_0 + u_1) \wedge (u_1 + u_2)
= u_0 \wedge u_1 + u_0 \wedge u_2 + u_1 \wedge u_2
\]

\[
u_3 = -(u_0 + u_1 + u_2)
\]

\[
\underbrace{u_0 \wedge u_1} + u_2 \wedge \underbrace{(u_0 + u_1)}
= \underbrace{u_1 \wedge u_2} + u_0 \wedge \underbrace{(u_1 + u_2)}
\]

\[
u_2 \wedge \underbrace{(u_3 + u_0 + u_1)}_{-u_2} = 0 \qquad \text{ok}
\]

\[
\boxed{\lambda_0 = 0} \Rightarrow u_1 u_{0i} = \lambda_i u_{i-1} u_i
\]

\[
\begin{array}{lll}
s_0 = (0, 0) & & \\
s_1 = (1, 0) & & \\
s_2 = (1, 1) & & (y_2 = 1) \\
s_3 = (\beta_1, \beta_1) & \beta_1 \neq 1, 0 & \\
s_4 = (\beta_1, y_4) & y_4 \neq \beta_1 & \\
s_5 = (\beta_2 \beta_1, \beta_2 y_4) & \beta_2 \neq 1, 0 & \\
s_6 = (\beta_2 \beta_1, y_6) & y_6 \neq \beta_2 y_4 & \\
s_7 = (\beta_3 \beta_2 \beta_1, \beta_3 y_6) & \beta_3 \neq 1, 0 & \\
s_8 = (\beta_3 \beta_2 \beta_1, y_8) & y_8 \neq \beta_3 y_6 & \\
{[s_9 = s_0]} & [\beta_4 = 0] &
\end{array}
\qquad
\begin{array}{l}
\beta_1\ \beta_2\ \beta_3 \\
y_4\ y_6\ y_8
\end{array}
\]

\[
\lambda_i u_{i-1} - \lambda_{i+1} u_i - \lambda_{i+1} u_{i+1}
+ \lambda_{i+2} u_{i+2} = 0
\]

\[
\begin{array}{ll}
\lambda_1 u_0 - \lambda_2 u_1 - \lambda_2 u_2 + \lambda_3 u_3 = 0 &
\lambda_3 u_3 = -\lambda_1 u_0 + \lambda_2 u_1 + \lambda_2 u_2 \\
\lambda_2 u_1 - \lambda_3 u_2 - \lambda_3 u_3 + \lambda_4 u_4 = 0 &
\lambda_4 u_4 = -\lambda_1 u_0 + \ldots \\
\lambda_3 u_2 - \lambda_4 u_3 - \lambda_4 u_4 + \lambda_5 u_5 = 0 &
\lambda_3 \lambda_5 u_5 = \\
\lambda_4 u_3 - \lambda_5 u_4 - \lambda_5 u_5 + \lambda_6 u_6 &
\end{array}
\]

\[
\begin{aligned}
\lambda_3 \lambda_5 u_5 &= \lambda_3 \bigl[ -\lambda_1 u_0
 + (\lambda_2 + \lambda_3)\, u_2 \bigr]
 + \lambda_4 (-\lambda_1 u_0 + \lambda_2 u_1 + \lambda_2 u_2)
 - \lambda_3^2 u_2 \quad \cdots \\
 &= \bigl( -\lambda_1 (\lambda_3 + \lambda_4) \bigr) u_0
 + \lambda_2 \lambda_4 u_1 + \lambda_2 (\lambda_3 + \lambda_4)\, u_2
\end{aligned}
\]

\[
\begin{array}{ll}
s_0 = (0, 0) & u_0 = (1, 0) \\
s_1 = (1, 0) & u_1 = (x - 1, y) \\
{[s_4 = (0, f_0)]} & u_2 = (x' - x, y' - y) \\
s_2 = (x, y) & u_3 = (-x', f_0 - y') \\
s_3 = (x', y') & u_4 = (0, -f_0)
\end{array}
\qquad
\begin{aligned}
f_1 &= y \\
f_2 &= (x - 1)(y' - y) - y(x' - x) \\
 &= (xy' - yx') - (y' - y) \\
f_3 &= (x' - x)(f_0 - y') + x'(y' - y) \\
 &= (xy' - yx') + f_0 (x' - x) \\
f_4 &= f_0 x' \\
f_0 &= f_0
\end{aligned}
\]

\[
\left\lbrace
\begin{aligned}
y &= f_1 \\
x' &= f_4/f_0 \\
xy' - y' &= f_2 + f_1 f_4/f_0 - f_1 \\
xy' - f_0 x &= f_3 + f_1 f_4/f_0 - f_4
\end{aligned}
\right.
\qquad\qquad
f_0 x - y' = -f_1 + f_2 - f_3 + f_4
\]

\[
\left\lbrace
\begin{aligned}
y &= f_1 \\
x' &= f_4/f_0 \\
y' &= f_0 x + X_1^4
\end{aligned}
\right.
\qquad
X_1^4 = f_1 - f_2 + f_3 - f_4 \qquad
X = f_0 - f_1 + f_2 - f_3 + f_4 = f_0 - X_1^4
\]

\[
x (f_0 x + X_1^4) - f_0 x - X_1^4 = f_2 + f_1 f_4/f_0 - f_1
\]

\[
f_0 x^2 + \underbrace{(X_1^4 - f_0)}_{-X} x
- (f_1 - f_2 + f_3 - f_4 + f_2 + f_1 f_4/f_0 - f_1)
\]

\[
f_0 x^2 - X x + (f_4 - f_3 - f_1 f_4/f_0) \qquad\qquad X = 1 - X_1^4
\]

\[
x = \frac{1}{2 f_0} \Bigl( X \pm \sqrt{X^2 - 4 f_0 f_4 + 4 f_0 f_3
+ 4 f_1 f_4} \Bigr)
\]

\[
\begin{aligned}
X^2 = f_0^2 + f_1^2 + f_2^2 + f_3^2 + f_4^2
 &- 2 f_0 f_1 + 2 f_0 f_2 - 2 f_0 f_3 + 2 f_0 f_4 \\
 &- 2 f_1 f_2 + 2 f_1 f_3 - 2 f_1 f_4 \\
 &- 2 f_2 f_3 + 2 f_2 f_4 \\
 &- 2 f_3 f_4
\end{aligned}
\]

\[
\Delta = f_0^2 + f_1^2 + f_2^2 + f_3^2 + f_4^2 + \ldots
\]

\[
\begin{aligned}
&= \textstyle\sum f_i^2 - 2 \sum f_i f_{i+1} + 2 \sum f_i f_{i+2} \\
&= \textstyle\sum f_i^2 + 2 \sum f_{i+1} (f_i - f_{i+1}) \\
&\phantom{=}\ (f_0 - f_1 + f_2 + f_3 - f_4)^2
 + 4 (f_1 f_3 + f_2 f_3 + f_2 f_4)
\end{aligned}
\]

\[
u_3 \wedge u_4 - u_2 \wedge u_3 = u_3 \wedge u_4 + u_3 \wedge u_2
= u_3 \wedge (u_2 + u_4) \qquad
(u_4 \wedge u_0)\bigl(u_3 \wedge (u_2 + u_4)\bigr) -
\]

\[
\begin{aligned}
&x^2 + y^2 + z^2 + 2yz + 2zx - 2xy = \\
&\xi^2 + \eta^2 - 2\xi\eta + 2(\xi + \eta) + 1 \\
&\underbrace{(\xi - \eta)}_{U}{}^2 + 2\underbrace{(\xi + \eta)}_{V} + 1
\end{aligned}
\]

\[
\boxed{\lambda_{i-1} u_{i-2} - \lambda_i (u_{i-1} + u_i) + \lambda_{i+1}
u_{i+1} = 0}
\]

\[
\underbrace{u_0 \quad u_1}_{\text{base}} \qquad u_2 = \xi u_0 + \eta u_1
\]

\[
\begin{aligned}
u_3 &= \frac{1}{\lambda_3} \bigl( \lambda_2 (u_1 + u_2) - \lambda_1 u_0 \bigr) \\
u_4 &= \frac{1}{\lambda_3 \lambda_4} (\quad \\
u_5 &= \frac{1}{\lambda_3 \lambda_4 \lambda_5} (\quad \text{------} \quad)
\end{aligned}
\]

\[
u_{n-1} = \frac{\lambda_{n-2}}{\lambda_3 \cdots \lambda_{n-1}}
\bigl( P_{n-1} u_0 + Q_{n-1} u_1 + R_{n-1} u_2 \bigr)
\]

\[
u_i = \frac{1}{\lambda_3 \lambda_4 \cdots \lambda_i}\, P_i(\lambda_1, \ldots,
\lambda_{i-1})\, u_0 + Q_i(\lambda_1, \ldots
\]

\[
u_3 = \lambda_3^{-1} \, (P_3 u_0 + Q_3 u_1 + R_3 u_2)
\]

\[
u_i = \frac{\lambda_{i-1}}{\lambda_3 \lambda_4 \cdots \lambda_i}
\bigl( P_i(\lambda_1, \ldots, \lambda_{i-1})\, u_0
 + Q_i(\lambda_1, \ldots, \lambda_{i-1})\, u_1
 + R_i(\lambda_1, \ldots, \lambda_{i-1})\, u_2 \bigr)
\qquad (i \geqslant 4)
\]

\begin{align*}
\lambda_{i+1} u_{i+1} &= \lambda_i (u_i + u_{i-1}) - \lambda_{i-1} u_{i-2} \\
 &= \frac{1}{\lambda_3 \cdots \lambda_{i-2}}
    \bigl[ P_i u_0 + Q_i u_1 + R_i u_2 \bigr]
  + \frac{\lambda_i \lambda_{i-2}}{\lambda_3 \cdots \lambda_{i-1}}
    \bigl[ P_{i-1} u_0 + Q_{i-1} u_1 + R_{i-1} u_2 \bigr] \\
 &\quad - \frac{\lambda_{i-1} \lambda_{i-3}}{\lambda_3 \lambda_4 \cdots \lambda_{i-2}}
    \bigl[ P_{i-2} u_0 + Q_{i-2} u_1 + R_{i-2} u_2 \bigr]
\end{align*}

\[
u_{i+1} = \frac{\lambda_i}{\lambda_3 \cdots \lambda_{i+1}}
\Bigl\lbrace \lambda_{i-1} \bigl[\quad\bigr]_1
 + \lambda_i \lambda_{i-2} \bigl[\quad\bigr]_2
 - \lambda_{i-1}^2 \lambda_{i-3} \bigl[\quad\bigr]_3 \Bigr\rbrace
\]

\[
\begin{array}{ccccc}
i-3 & i-2 & i-1 & i & i+1 \\
0 & -1 & 1 & 0 & -1
\end{array}
\]

\[
\left\lbrace
\begin{array}{l}
P_{i+1} = \lambda_{i-1} P_i + \lambda_i \lambda_{i-2} P_{i-1}
 - \lambda_{i-1}^2 \lambda_{i-3} P_{i-2} \\
Q_{i+1} = \lambda_{i-1} Q_i + \lambda_i \lambda_{i-2} Q_{i-1}
 - \lambda_{i-1}^2 \lambda_{i-3} Q_{i-2} \\
R_{i+1} = \lambda_{i-1} R_i + \lambda_i \lambda_{i-2} R_{i-1}
 - \lambda_{i-1}^2 \lambda_{i-3} R_{i-2}
\end{array}
\right.
\qquad i \geqslant 6
\]

\[
P_3 = -\lambda_1, \qquad Q_3 = \lambda_2, \qquad R_3 = \lambda_2
\]

\[
(\lambda_3 \cdots \lambda_{n-1}) \sum_{0}^{n-1} u_i
 = \Bigl( \sum_{i=0}^{n-1} P_i(\lambda_1, \ldots, \lambda_{n-2}) \Bigr) u_0
 + (\quad) u_1 + (\quad) u_2
\]

\[
\widetilde{P}_{n-1}(\lambda_1, \ldots, \lambda_{n-2})\, u_0
 + \widetilde{Q}_{n-1}(\lambda_1, \ldots, \lambda_{n-2})\, u_1
 + \widetilde{R}_{n-1}(\lambda_1, \ldots, \lambda_{n-2})\, u_2
\]

\begin{align*}
\widetilde{P}_{n-1}(\lambda_1, \ldots, \lambda_{n-1})
 = (\lambda_3 \cdots \lambda_{n-1})
 &+ \bigl( \lambda_4 \lambda_5 \cdots \lambda_{n-1} P_3
 + \lambda_3 \lambda_5 \lambda_6 \cdots \lambda_{n-1} P_4 \\
 &\qquad + \lambda_4 \lambda_6 \lambda_7 \cdots \lambda_{n-1} P_5 + \cdots \bigr)
\end{align*}

\[
\widetilde{P}_{n-1} \in \mathbf{Z}[\Lambda_1, \ldots, \Lambda_{n-1}]
\]

\[
(0) \qquad \widetilde{P}_{n-1} u_0 + \widetilde{Q}_{n-1} u_1
 + \widetilde{R}_{n-1} u_2 = 0
\]

\[
\underset{\textstyle u_0}{\underset{\shortparallel}{u_n}}
 = \frac{\lambda_{n-1}}{\lambda_3 \cdots \lambda_{n-1} \lambda_n}
 \bigl[ P_n(\lambda_1, \ldots, \lambda_{n-1}, \lambda_n)\, u_0
 + Q_n(\lambda_1, \ldots, \lambda_n)\, u_1
 + R_n(\lambda_1, \ldots, \lambda_n)\, u_2 \bigr]
\]

\[
(1) \qquad \bigl[ \lambda_{n-1} P_n - (\lambda_3 \cdots \lambda_{n-1} \lambda_n) \bigr] u_0
 + \lambda_{n-1} Q_n(\quad)\, u_1 + \lambda_{n-1} R_n(\quad)\, u_2 = 0
 \qquad (\lambda_n = \lambda_0)
\]

\[
\underset{\textstyle u_1}{\underset{\shortparallel}{u_{n+1}}}
 = \frac{\lambda_n}{\lambda_3 \cdots \lambda_{n+1}}
 \bigl[ P_{n+1}(\lambda_1, \ldots, \lambda_{n-1}, \lambda_n, \lambda_{n+1})\, u_0
 + \cdots \bigr]
 \qquad (\lambda_n = \lambda_0,\ \lambda_{n+1} = \lambda_1)
\]

\[
(2) \qquad \lambda_n P_{n+1}(\quad)\, u_0
 + \bigl( \lambda_n Q_{n+1} - (\lambda_3 \lambda_4 \cdots \lambda_{n+1}) \bigr) u_1
 + \lambda_n R_{n+1}(\quad)\, u_2 = 0
\]

\[
u_{n+2} = \frac{\lambda_{n+1}}{\lambda_3 \cdots \lambda_{n+2}}
\bigl[ P_{n+2} u_0 + Q_{n+2} u_1 + R_{n+2} u_2 \bigr] = u_2
\]

\[
(3) \qquad \lambda_{n+1} P_{n+2}(\quad)\, u_0 + \lambda_{n+1} Q_{n+2}(\quad)\, u_1
 + \lambda_{n+1} R_{n+2}(-)(\lambda_3 \cdots \lambda_{n+2})\, u_2 = 0
\]

\[
\left\lbrace
\begin{array}{l}
A_0^n(\lambda_1, \ldots, \lambda_{n-2})\, u_0 + B_0^n(\lambda_1, \ldots, \lambda_{n-2})\, u_1
 + C_0^n(\lambda_1, \ldots, \lambda_{n-2})\, u_2 = 0 \\
A_1^n(\lambda_0, \ldots, \lambda_{n-1})\, u_0 + B_1^n(\lambda_0, \ldots, \lambda_{n-1})\, u_1
 + C_1^n(\lambda_0, \ldots, \lambda_{n-1})\, u_2 = 0 \\
A_2^n(\quad)\, u_0 + B_2^n(\quad)\, u_1 + C_2^n(\quad)\, u_2 = 0 \\
A_3^n(\quad)\, u_0 + B_3^n(\quad)\, u_1 + C_3^n(\quad)\, u_2 = 0
\end{array}
\right.
\]

\[
\begin{array}{c}
A_i^n,\ B_i^n,\ C_i^n \in \mathbf{Z}[\Lambda_0, \ldots, \Lambda_{n-1}] \\
\cup \\
A_0^n,\ B_0^n,\ C_0^n \in \mathbf{Z}[\Lambda_1, \ldots, \Lambda_{n-2}]
\end{array}
\]

\[
\lambda_1 u_0 - \lambda_2 (u_1 + u_2) + \lambda_3 u_3 = 0
\]

\[
\begin{array}{ll}
u_1 + u_2 = -\alpha_0 u_0 & \\
u_3 = \beta u_0 &
\end{array}
\qquad
\left\lbrace
\begin{array}{l}
u_2 = -\alpha u_0 - u_1 \\
u_3 = \beta u_0
\end{array}
\right.
\]

\[
\left\lbrace
\begin{array}{l}
u_3 = \alpha u_0 + (1 + \gamma) u_1 \\
u_4 = \delta u_1
\end{array}
\right.
\qquad
\begin{array}{l}
\gamma = -1 \\
\beta = \alpha
\end{array}
\]

\[
u_2 = \alpha u_0 - u_1, \qquad u_3 = \alpha u_0, \qquad u_4 = \delta u_1
\]

\[
u_{i+3} = \alpha_i u_i
\]

\[
u_0 \ / \ u_1 \ /
\]

\[
\begin{array}{l}
u_2 = -\alpha_0 u_0 - u_1 \\
u_3 = \alpha_0 u_0 \\
u_4 = \alpha_1 u_1 \\
u_5 = \alpha_2 (\alpha_0 u_0 - u_1) \\
u_6 = \alpha_3 \alpha_0 u_0 \\
u_7 = \alpha_4 \alpha_2 (-\alpha_0 u_0 - u_1) \\
u_8 = \alpha_5 \alpha_3 \alpha_0 u_0 \\
u_9 = \alpha_6 \alpha_4 \alpha_2 (-\alpha_0 u_0 - u_1)
\end{array}
\]

\[
\left.
\begin{array}{l} u_0 \\ u_1 \end{array}
\right\rbrace \text{ lin.\ indép.}
\]

\[
\beta_1 = \alpha_1, \qquad \beta_2 = \alpha_2, \qquad \beta_3 = \alpha_3 \alpha_0
\]

\[
\begin{array}{l}
u_2 = -\beta_0 u_0 - u_1 \\
u_3 = \beta_0 u_0 \\
u_4 = \beta_1 u_1 \\
u_5 = \beta_2 u_2 \\
u_6 = \beta_3 u_0 \\
u_7 = \beta_4 u_2
\end{array}
\]

\[
u_3 + u_4 = \beta_0 u_0 + \beta_1 u_1 \parallel u_2 = -\beta_0 u_0 - u_1
\qquad \beta_1 = 1 \quad \text{i.e.} \quad u_1 = u_4
\]

\[
\begin{array}{cccccccccc}
u_0 & u_1 & u_2 & u_0 & u_1 & u_2 & u_0 & u_1 & u_2 & \ldots \\
 & & & \shortparallel & \shortparallel & \shortparallel & \shortparallel & \shortparallel & \shortparallel & \\
 & & & u_3 & u_4 & u_5 & u_6 & u_7 & u_8 &
\end{array}
\]

\[
\begin{array}{l}
u_1 + u_2 \parallel u_0 \\
u_2 + u_0 \parallel u_1 \\
u_0 + u_1 \parallel u_2
\end{array}
\qquad
\begin{array}{l}
u_2 = \alpha_0 u_0 - u_1 \\
u_2 + u_0 = (1 + \alpha_0) u_0 - u_1 \parallel u_1
 \quad \text{i.e.} \quad 1 + \alpha_0 = 0, \quad \alpha_0 = -1 \\
u_2 = -u_0 - u_1
\end{array}
\]

\[
(*) \qquad \lambda_i u_{i-1} - \lambda_{i+1} (u_i + u_{i+1}) + \lambda_{i+2} u_{i+2} = 0
\qquad \forall i \in \mathbf{Z}/n\mathbf{Z}
\]

\[
\left\lbrace
\begin{array}{l}
u_{i+3} = u_i \\
u_{i+1} + u_{i+2} = -u_i \qquad \forall i
\end{array}
\right.
\]

\[
\left\lbrace
\begin{array}{l}
\forall i \quad u_i = u_{i+3} \\
u_0 + u_1 + u_2 = 0
\end{array}
\right.
\]

\[
\begin{array}{ccc}
N(u_*) & \longrightarrow & k^3 \\
(\lambda_i) & \longmapsto & (\lambda_1, \lambda_2, \lambda_3)
\end{array}
\]

\[
1 \leqslant \dim N(u_*) \leqslant 3 .
\]

\[
N(u_*) \simeq \operatorname{Im}\bigl( N(u_*) \to k^3 \bigr)
 \subset \bigl\lbrace (\lambda_1, \lambda_2, \lambda_3) \bigm|
 \lambda_1 u_0 - \lambda_2 (u_1 + u_2) + \lambda_3 u_3 = 0 \bigr\rbrace
\]

\[
= \operatorname{Ker}\bigl( (\lambda_1, \lambda_2, \lambda_3) \mapsto
 \lambda_1 u_0 - \lambda_2 (u_1 + u_2) + \lambda_3 u_3 \bigr)
\]

\[
\dim N(u_*) \leqslant \dim \operatorname{Ker}
 = 3 - \operatorname{rg}(u_0, u_1 + u_2, u_3)
 = \left\lbrace
 \begin{array}{l}
 1 \text{ si } \operatorname{rg} = 2 \\
 2 \text{ si } \operatorname{rg} = 1
 \end{array}
 \right.
\]

\[
\lambda_i u_{i-1} - \lambda_{i+1} (\underbrace{u_i + u_{i+1}}_{-u_{i-1}})
 + \underset{\textstyle \lambda_{i-1}}{\underset{\shortparallel}{\lambda_{i+2}}}
 (\underset{\textstyle u_{i-1}}{\underset{\shortparallel}{u_{i+2}}}) = 0
 \qquad \forall i
\]

\[
(\lambda_i + \lambda_{i-1} + \lambda_{i+1})\, u_{i-1} = 0
\]

\[
N(u_*) = \bigl\lbrace (\lambda_i) \in k^{\mathbf{Z}/n\mathbf{Z}} \bigm|
 \lambda_i u_{i-1} - \lambda_{i+1} (u_i + u_{i+1}) + \lambda_{i+2} u_{i+2} = 0
 \quad \forall i \in \mathbf{Z}/n\mathbf{Z} \bigr\rbrace
\]

\[
N(u_*) \not\subset \bigcup H_i \qquad
\bigl( \Longrightarrow \dim N(u_*) \in \lbrace 1, 2 \rbrace \bigr)
\]

\[
\left\lbrace
\begin{array}{l}
\text{a) } \sum u_i = 0 \\
\text{b) } \forall\, 1 \leqslant i \leqslant n-1, \text{ on ait }
 u_{i-1}, u_i \parallel u_1, u_{0i}
\end{array}
\right.
\]

\[
u_1 u_{0i} = \lambda_i\, u_i u_{i-1}
\]