Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory, extending for example classical classes like P or PSPACE to operators in Analysis [doi:10.1137/S0097539794263452, doi:10.1145/2189778.2189780]. The degree subclassifies ordinary polynomial growth into linear, quadratic, cubic etc. In order to similarly classify second-order polynomials, define their degree to be an 'arctic' first-order polynomial (namely a term/expression over variable $D$ and operations $+$ and $\cdot$ and $\max$). This degree turns out to transform as nicely under (now two kinds of) polynomial composition as the ordinary one. We also establish a normal form and semantic uniqueness for second-order polynomials. Then we define the degree of a third-order polynomial to be an arctic second-order polynomial, and establish its transformation under three kinds of composition.

翻译：二阶多项式通过允许变量取值范围从数值扩展至函数，从而推广了经典的一阶多项式。其研究动机源于高阶计算复杂性理论中的应用，例如将经典复杂度类（如P或PSPACE）扩展为分析学中的算子[doi:10.1137/S0097539794263452, doi:10.1145/2189778.2189780]。度数将普通多项式增长细分为线性、二次、三次等类型。为对二阶多项式进行类似分类，定义其度数为“北极”一阶多项式（即关于变量$D$、运算$+$、$\cdot$和$\max$的项/表达式）。该度数在（现含两种类型的）多项式复合下展现出与普通度数同样良好的变换性质。我们建立了二阶多项式的范式与语义唯一性。进一步地，定义三阶多项式的度数为“北极”二阶多项式，并建立其在三种复合类型下的变换规律。