Saturday, December 28, 2013

Degree of a transseries

I believe that the construction of surreal numbers and transseries should be done through a process of extensions going in order from natural numbers, to integers, to rationals, to reals, and then to real polynomials. In keeping with the fact that these number systems are extensions of one another we can compute the integer part of a real number or the standard part of a real number.

An analogous notion is computing the degree of a transseries. The way I think we should go about doing this is to compute the standard part of $\frac{ln(f(x))}{ln(x)}$. Since transseries are closed under logs and division this remains a transseries for any transseries so we can still compute its standard part.

If we plug in a constant we get $\frac{ln(c)}{ln(x)}$ which goes to zero for any constant. if we plug in $ln(x)$ we get $\frac{ln(ln(x))}{ln(x)}$ which also goes to zero, If we plug in the square root we get $\frac{ln(\sqrt{x})}{ln(x)}$ which goes to 1/2, if we plug in quadratics we get a degree of two, if we plug in quintics we get a degree of three, and if we go on to plug in exponentials we get a degree of infinity. It is necessary to use the infinity to deal with exponential degrees just as it is necessary to use infinity to deal with transfinite values when computing the standard part.

The next phase to build our transseries system is to make use of exponential and logarithmic values. With exponentials we get growth rates greater then any polynomial degree and with logarithms we get growth rates that are so small they are not distinguishable in the degrees system. In essence exponentials are of infinite degree and logarithms are of infinitesimal degree.

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