For a finite set $\cal F$ of polynomials over fixed finite prime field of size $p$ containing all polynomials $x^2 - x$ a Nullstellensatz proof of the unsolvability of the system $$ f = 0\ ,\ \mbox{ all } f \in {\cal F} $$ in the field is a linear combination $\sum_{f \in {\cal F}} \ h_f \cdot f$ that equals to $1$ in the ring of polynomails. The measure of complexity of such a proof is its degree: $\max_f deg(h_f f)$. We study the problem to establish degree lower bounds for some {\em extended} NS proof systems: these systems prove the unsolvability of $\cal F$ by proving the unsolvability of a bigger set ${\cal F}\cup {\cal E}$, where set $\cal E$ may use new variables $r$ and contains all polynomials $r^p - r$, and satisfies the following soundness condition: -- - Any $0,1$-assignment $\overline a$ to variables $\overline x$ can be appended by an assignment $\overline b$ to variables $\overline r$ such that for all $g \in {\cal E}$ it holds that $g(\overline a, \overline b) = 0$. We define a notion of pseudo-solutions of $\cal F$ and prove that the existence of pseudo-solutions with suitable parameters implies lower bounds for two extended NS proof systems ENS and UENS defined in Buss et al. (1996/97). Further we give a combinatorial example of $\cal F$ and candidate pseudo-solutions based on the pigeonhole principle.

翻译：对于固定有限素域（大小为$p$）上的有限多项式集合$\cal F$，若该集合包含所有多项式$x^2 - x$，则在域中证明方程组 $$ f = 0\ ,\ \mbox{ 对所有 } f \in {\cal F} $$ 无解的Nullstellensatz证明，是多项式环中等于$1$的线性组合 $\sum_{f \in {\cal F}} \ h_f \cdot f$。此类证明的复杂度度量是其度数：$\max_f deg(h_f f)$。我们研究为某些**扩展**NS证明系统建立度数下界的问题：这些系统通过证明更大集合${\cal F}\cup {\cal E}$的无解性来证明$\cal F$的无解性，其中集合$\cal E$可使用新变量$r$并包含所有多项式$r^p - r$，且满足以下可靠性条件：——任意对变量$\overline x$的$0,1$-赋值$\overline a$均可扩展为对变量$\overline r$的赋值$\overline b$，使得对所有$g \in {\cal E}$均有$g(\overline a, \overline b) = 0$。我们定义了$\cal F$的伪解概念，并证明具有适当参数的伪解存在性可引出Buss等人（1996/97）定义的两种扩展NS证明系统ENS与UENS的下界。进一步，我们基于鸽巢原理给出了$\cal F$及候选伪解的组合示例。