Assuming that the differential field (K,δ) is differentially large, in the sense of [León Sánchez and Tressl, Differentially large fields, preprint (2020); arXiv:2005.00888], and “bounded” as a field, we prove that for any linear differential algebraic group G over K, the differential Galois (or constrained) cohomology set Hδ1(K,G) is finite. This applies, among other things, to closed ordered differential fields in the sense of [Singer, The model theory of ordered differential fields, J. Symb. Logic 43(1) (1978) 82–91], and to closedp-adic differential fields in the sense of [Tressl, The uniform companion for large differential fields of characteristic 0, Trans. Amer. Math. Soc. 357(10) (2005) 3933–3951]. As an application, we prove a general existence result for parameterized Picard–Vessiot (PPV) extensions within certain families of fields; if (K,δx,δt) is a field with two commuting derivations, and δxZ = AZ is a parameterized linear differential equation over K, and (Kδx,δt) is “differentially large” and Kδx is bounded, and (Kδx,δt) is existentially closed in (K,δt), then there is a PPV extension (L,δx,δt) of K for the equation such that (Kδx,δt) is existentially closed in (L,δt). For instance, it follows that if the δx-constants of a formally real differential field (K,δx,δt) is a closed orderedδt-field, then for any homogeneous linear δx-equation over K there exists a PPV extension that is formally real. Similar observations apply to p-adic fields.

中文翻译：

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微分伽罗瓦上同调和参数化 Picard-Vessiot 扩展

假设差分场(ķ,δ)在 [León Sánchez 和 Tressl，Differentially large fields，preprint (2020) 的意义上，差异很大；arXiv:2005.00888]，并将“有界”作为一个域，我们证明对于任何线性微分代数群G超过ķ，微分伽罗瓦（或约束）上同调集Hδ1(ķ,G)是有限的。除其他外，这适用于闭序微分场在[辛格，有序微分场的模型理论，J. 辛布。逻辑 43(1) (1978) 82–91]，以及关闭p-adic 微分场在 [Tressl，特征的大微分场的均匀伴侣0,反式。阿米尔。数学。社会党。 357(10) (2005) 3933–3951]。作为一个应用，我们证明了参数化 Picard-Vessiot (PPV) 扩展在某些域族中的普遍存在结果；如果(ķ,δX,δ吨)是具有两个通勤派生的字段，并且δXZ = 一种Z是一个参数化的线性微分方程ķ， 和(ķδX,δ吨)是“差异很大”并且ķδX是有界的，并且(ķδX,δ吨)存在性地封闭于(ķ,δ吨), 那么有一个 PPV 扩展(大号,δX,δ吨)的ķ对于方程，使得(ķδX,δ吨)存在性地封闭于(大号,δ吨). 例如，如果δX-形式实微分场的常数(ķ,δX,δ吨)是一个封闭有序δ吨-场地, 那么对于任何齐次线性δX-方程结束ķ存在形式上真实的 PPV 扩展。类似的观察适用于p-adic 领域。