标题： 高阶多项式的次数 显示英文标题

标题： Degrees of Second and Higher-Order Polynomials

摘要： 二阶多项式通过允许额外的变量范围函数而不是值，推广了经典的的一阶多项式。 我们受到它们在高阶计算复杂性理论中的应用的启发，例如将经典的类如P或PSPACE扩展为分析中的算子 [doi:10.1137/S0097539794263452, doi:10.1145/2189778.2189780]。 次数将普通的多项式增长细分为线性、二次、三次等。 为了以类似的方式对二阶多项式进行分类，将其次数定义为一个“极地”一阶多项式（即变量$D$和运算$+$、$\cdot$和$\max$的项/表达式）。 这个次数在（现在有两种类型的）多项式复合下表现出与普通次数同样良好的转换特性。 我们还建立了二阶多项式的规范形式和语义唯一性。 然后我们将三阶多项式的次数定义为一个极地二阶多项式，并建立其在三种类型的复合下的转换特性。 显示英文摘要

摘要： 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.