Degrees of Second and Higher-Order Polynomials

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.