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Given a matrix $A$ and vector $b$ with polynomial entries in $d$ realvariables $\delta=(\delta_1,\ldots,\delta_d)$ we consider the following notionof feasibility: the pair $(A,b)$ is locally feasible if there exists an openneighborhood $U$ of $0$ such that for every $\delta\in U$ there exists $x$satisfying $A(\delta)x\ge b(\delta)$ entry-wise. For $d=1$ we construct apolynomial time algorithm for deciding local feasibility. For $d \ge 2$ we showlocal feasibility is NP-hard. This also gives the first polynomial-timealgorithm for the asymptotic linear program problem introduced by Jeroslow in1973.  As an application (which was the primary motivation for this work) we give acomputer-assisted proof of ergodicity of the following elementary 1D cellularautomaton: given the current state $\eta_t \in \{0,1\}^{\mathbb{Z}}$ the nextstate $\eta_{t+1}(n)$ at each vertex $n\in \mathbb{Z}$ is obtained by$\eta_{t+1}(n)= \text{NAND}\big(\text{BSC}_\delta(\eta_t(n-1)),\text{BSC}_\delta(\eta_t(n))\big)$. Here the binary symmetric channel$\text{BSC}_\delta$ takes a bit as input and flips it with probability $\delta$(and leaves it unchanged with probability $1-\delta$). It is shown that thereexists $\delta_0>0$ such that for all $0<\delta<\delta_0$ the distribution of$\eta_t$ converges to a unique stationary measure irrespective of the initialcondition $\eta_0$.  We also consider the problem of broadcasting information on the 2D-grid ofnoisy binary-symmetric channels $\text{BSC}_\delta$, where each node may applyan arbitrary processing function to its input bits. We prove that there exists$\delta_0'>0$ such that for all noise levels $0<\delta<\delta_0'$ it isimpossible to broadcast information for any processing function, as conjecturedby Makur, Mossel and Polyanskiy.

Linear Programs with Polynomial Coefficients and Applications to 1D Cellular Automata