二阶及高阶多项式的次数
Degrees of Second and Higher-Order Polynomials
AI总结：
本文研究二阶及高阶多项式的次数分类，将其次数定义为“极地”多项式，证明其在多项式复合下变换良好，并确立了范式、语义唯一性及三阶多项式次数的变换规律。
摘要：
二阶多项式推广了经典的一阶多项式，允许使用在函数而非值上取值的附加变量。我们的动机源于它们在高阶计算复杂性理论中的应用，例如将 P 或 PSPACE 等经典复杂性类扩展至分析中的算子 [doi:10.1137/S0097539794263452, doi:10.1145/2189778.2189780]。次数将普通多项式增长细分为线性、二次、三次等。为了类似地对二阶多项式进行分类，我们将其次数定义为一阶“极地”多项式（即在变量 $D$ 以及运算 $+$、$\cdot$ 和 $\max$ 上的项/表达式）。结果表明，该次数在（现有两种）多项式复合下的变换与普通次数一样良好。我们还为二阶多项式确立了范式与语义唯一性。随后，我们将三阶多项式的次数定义为极地二阶多项式，并确立了其在三种复合下的变换。
英文摘要：
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.
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