AI 中文总结
研究代数拟阵识别问题，通过关联射影平面，利用赫鲁绍夫斯基 - 齐尔伯定理等，构造从\(\mathbb{F}_p(x)\)上丢番图方程可解性到拟阵代数性的归约，证明其识别问题不可判定。
AI 中文摘要
我们证明了代数拟阵的识别问题是不可判定的。具体而言，不存在这样一种算法，它以有限集\(S\)和函数\(r\colon\mathcal{P}(S) \to \mathbb{Z}_{\ge 0}\)为输入，能判定是否存在一对域\(F \subset K\)以及函数\(f\colon S \to K\)，使得对所有\(A \subseteq S\)，\(\mathrm{ this http URL }_{K/F}(f(A)) = r(A)\)。已知当所涉域的特征被限制为零时该问题是可判定的。我们证明当特征未指定（即接受任何特征下的实现）或固定为素数\(p\)时是不可判定的。证明依赖于赫鲁绍夫斯基 - 齐尔伯的群配置定理以及埃文斯和赫鲁绍夫斯基关于“代数闭域中的射影平面”的工作。我们关联了两个不同的射影平面，最终构造了从\(\mathbb{F}_p(x)\)（\(p\)为素数）上丢番图方程的可解性到拟阵代数性的归约。对于所有\(p > 2\)，菲达斯证明了\(\mathbb{F}_p(x)\)上丢番图方程的可解性是不可判定的，后来维德拉证明了\(p = 2\)时也是如此。我们证明的核心部分是所谓域配置定理的一个变体。
英文摘要
We prove that the recognition problem for algebraic matroids is undecidable. Explicitly, this means that there is no algorithm that takes as input a finite set $S$ and a function $r\colon\mathcal{P}(S) \to \mathbb{Z}_{\ge 0}$ (where $\mathcal{P}(S)$ is the power set) and decides whether there exists a pair of fields $F \subset K$, and a function $f\colon S \to K$, such that for all $A \subseteq S$: $\mathrm{trdeg}_F f(A) = r(A)$. This problem is known to be decidable if the characteristic of the fields involved is constrained to be zero. We prove that it is undecidable if the characteristic is either left unspecified (in which case a realization over any characteristic is accepted) or fixed to be a prime $p$. The proof relies on Hrushovski--Zilber's Group Configuration Theorem and on the work of Evans and Hrushovski on "Projective Planes in Algebraically Closed Fields". We relate two different such projective planes, and eventually construct a reduction from the solvability of Diophantine equations over $\mathbb{F}_p(x)$ ($p$ prime) to algebraicity of matroids. Solvability of Diophantine equations over $\mathbb{F}_p(x)$ was proved to be undecidable by Pheidas for all $p > 2$, and later by Videla for $p=2$. A central part of our proof is a variant of the so-called Field Configuration Theorem.
Comments29 pages, 7 figures. Comments are welcome! This version corrects the arxiv abstract and some latex issues, as well as a few typos
