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 $\max$). Our normal form and semantic uniqueness results for second-order polynomials assert said second-order degree to be well-defined; and it turns out to transform well under (now two kinds of) polynomial composition. More generally 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：二阶多项式是对经典一阶多项式的推广，允许使用取值于函数而非数值的附加变量。其研究动机源于在更高阶计算复杂性理论中的应用，例如将P或PSPACE等经典类扩展为分析中的算子[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。为类似地分类二阶多项式，将其度数定义为“北极”一阶多项式（即由变量$D$与运算$+$、$\cdot$、$\max$构成的项/表达式）。我们对二阶多项式的范式及语义唯一性结果保证了上述二阶度数的良好定义，且该度数在（现为两种类型的）多项式复合下呈现良好变换性质。更一般地，我们将三阶多项式的度数定义为北极二阶多项式，并建立其在三种类型复合下的变换法则。