We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in \(\mathbb {K}[x]\). The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type \(f(\alpha )=f(\beta )\) for \(\alpha ,\beta\) in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

我们为\(\mathbb {K}[x]\)中有限余维的子代数A引入子代数谱Sp ( A )的概念。频谱是基础场的有限子集。我们还介绍了一个工具，即A的特征多项式，它以频谱作为其零集。可以从A的生成元计算特征多项式，从而使我们能够找到由生成元给出的代数的谱。我们继续使用谱来获得有限余维次代数的描述。更准确地说，我们表明A可以用一组条件来描述，每个条件都是\(f(\alpha )=f(\beta )\)对于Sp ( A ) 中的\(\alpha ,\beta\)或声明在Sp ( A )的元素中评估的不同阶导数的某些线性组合等于零的类型。我们使用这些类型的条件，通过一个归纳过程，找到对多达三个余维数的子代数的明确描述。这些描述还包括每个子代数族的 SAGBI 基。