Incomplete Exponential Families and Unbiased Minimum Variance Estimates. I

Exponential family (9) of distributions on $R^1$ with polynomial relations (10) between the natural parameters $\vartheta_1,\dots,\vartheta_s$ is considered. The problem of unbiased estimation based on an independent sample of size $n\ge3$ from that population is investigated.
The main result of the paper foranulated as the basic theorem gives necessary and sufficient conditions for an arbitrary polynomial of sufficient statistics to be the best unbiased estimator of its expectation. This theorem solves one of the problems posed by Yu. V. Linnik in [3]. The original statistical problem is reduced (Lemma 2) to a differential-algebraic one by means of $D$-method due to Wijsman [7]. Some other results (Theorems 1 and 2) have an independent interest.