This Paper defines and explores solution to the problem of \emph{Inversion of a finite Sequence} over the binary field, that of finding a prefix element of the sequence which confirms with a \emph{Recurrence Relation} (RR) rule defined by a polynomial and satisfied by the sequence. The minimum number of variables (order) in a polynomial of a fixed degree defining RRs is termed as the \emph{Polynomial Complexity} of the sequence at that degree, while the minimum number of variables of such polynomials at a fixed degree which also result in a unique prefix to the sequence and maximum rank of the matrix of evaluation of its monomials, is called \emph{Polynomial Complexity of Inversion} at the chosen degree. Solutions of this problems discovers solutions to the problem of \emph{Local Inversion} of a map $F:\ftwo^n\rightarrow\ftwo^n$ at a point $y$ in $\ftwo^n$, that of solving for $x$ in $\ftwo^n$ from the equation $y=F(x)$. Local inversion of maps has important applications which provide value to this theory. In previous work it was shown that minimal order \emph{Linear Recurrence Relations} (LRR) satisfied by the sequence known as the \emph{Linear Complexity} (LC) of the sequence, gives a unique solution to the inversion when the sequence is a part of a periodic sequence. This paper explores extension of this theory for solving the inversion problem by considering \emph{Non-linear Recurrence Relations} defined by a polynomials of a fixed degree $>1$ and satisfied by the sequence. The minimal order of polynomials satisfied by a sequence is well known as non-linear complexity (defining a Feedback Shift Register of smallest order which determines the sequences by RRs) and called as \emph{Maximal Order Complexity} (MOC) of the sequence. However unlike the LC there is no unique polynomial recurrence relation at any degree.

翻译：本文定义并探讨了在二元域上\emph{有限序列求逆}问题的解决方案，即寻找序列的一个前缀元素，该元素符合由多项式定义且序列所满足的\emph{递推关系}规则。在固定次数下，定义递推关系的多项式所需的最小变量数（阶数）称为序列在该次数下的\emph{多项式复杂度}；而在固定次数下，此类多项式的最小变量数若同时能产生序列的唯一前缀且其单项式求值矩阵的秩达到最大，则称为选定次数下的\emph{求逆多项式复杂度}。该问题的解可用于解决映射$F:\ftwo^n\rightarrow\ftwo^n$在点$y$（属于$\ftwo^n$）的\emph{局部求逆}问题，即从方程$y=F(x)$中求解$\ftwo^n$中的$x$。映射的局部求逆具有重要应用价值，这提升了本理论的意义。先前的研究表明，由序列满足的最小阶\emph{线性递推关系}（即序列的\emph{线性复杂度}）在序列为周期序列的一部分时，能为求逆问题提供唯一解。本文通过考虑由固定次数$>1$的多项式定义且序列满足的\emph{非线性递推关系}，探索了该理论在解决求逆问题上的扩展。序列所满足多项式的最小阶数通常称为非线性复杂度（定义了通过递推关系确定序列的最小阶反馈移位寄存器），亦称为序列的\emph{最大阶复杂度}。然而与线性复杂度不同，在任何次数下均不存在唯一的非线性多项式递推关系。