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Consider the following system of quadratic polynomials $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,....,x_n]$ :

$f_1 (\bar{x}) = x_1 + x_n^2 + q_1$

$f_i(\bar{x}) = x_i + q_i$ for $i \in \{2,...,n-1 \}$

$f_n(\bar{x}) = g(\bar{x}) + l^2$

where $q_1,...,q_n$ are homogenous quadratics and $x_n^2, x_1^2 \notin supp(q_i)$ for $ i \in \{1,...,n-1\}$. Here $l$ is homogeous linear polynomial and $g(\bar{x}) = \sum_{i \neq j} a_{i,j} x_i x_j$.

Suppose for some $1 \leq k \leq n$, we know $f_n + A(f_1,...,f_{n-1}) = x_n^{2^k}$ mod $\langle \bar{x} \rangle^{2^k + 1}$, where $A \in \bar{\mathbb{F}}_2[y_1,....,y_{n-1}]$.

Can we say that $l \neq 0$?

Any help would be appreciated.

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  • $\begingroup$ The question is whether $ l \neq 0$ always $\endgroup$
    – Rishabh Kothary
    Commented Dec 26, 2023 at 5:55
  • $\begingroup$ And what is $\langle \bar{x} \rangle^{2^k + 1}$? $\endgroup$
    – Max Alekseyev
    Commented Dec 26, 2023 at 22:27
  • $\begingroup$ I meant the ideal $\langle x_1^{i_1},...,x_n^{i_n} | i_1 + .... + i_n = 2^k + 1 \rangle$. I am working in the precision upto degree $2^k$ $\endgroup$
    – Rishabh Kothary
    Commented Dec 29, 2023 at 9:07
  • $\begingroup$ What does it mean $x_n^2, x_1^2 \notin supp(q_i)$ ? $\endgroup$
    – Max Alekseyev
    Commented Jan 6 at 17:03

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