Given an affine algebra $R=P/I$, where $P=K[x_1,\dots ,x_n]$ is a polynomial ring over a field $K$ and $I$ is an ideal in $P$, we study re-embeddings of the affine scheme $\mathrm{\text{Spec}}(R)$, i.e. presentations $R\cong P^{\prime }/I^{\prime }$ such that $P^{\prime }$ is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials $f_i$ in the ideal $I$ which are coherently separating in the sense that they are of the form $f_i=z_i-g_i$ with an indeterminate $z_i$ which divides neither a term in the support of $g_i$ nor in the support of $f_j$ for $j\ne i$. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of $I$. The dimension of the cotangent space of $R$ at a $K$-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of $R$ and found an optimal re-embedding.

给定一个仿射代数$R=磷/我$， 在哪里$磷=ķ[X_1,\dots ,X_n]$是域上的多项式环$ķ$和$我$是一个理想的 $磷$，我们研究仿射方案的重新嵌入$\mathrm{\text{规格}}(R)$，即演示文稿$R\cong {磷}^{\text{'}}/{我}^{\text{'}}$这样${磷}^{\text{'}}$是较少不定项的多项式环。为了找到这样的重新嵌入，我们使用多项式$F_{\mathrm{一世}}$在理想中$我$它们是连贯地分开的，因为它们是形式$F_{\mathrm{一世}}=z_{\mathrm{一世}}-G_{\mathrm{一世}}$带有不确定的$z_{\mathrm{一世}}$它既不划分一个术语来支持$G_{\mathrm{一世}}$也不支持$F_j$为了$j\ne \mathrm{一世}$. 这样的多项式集合的可能数量由 Gröbner fan 控制$我$. 余切空间的维数 $R$在一个$ķ$-线性最大理想是嵌入维数的下界，如果我们找到对应于这个界的相干分离多项式，我们知道我们已经确定了嵌入维数$R$并找到了一个最佳的重新嵌入。