In the lecture on Monday, April 20 we will skip a calculation. Below is the calculation in its gory details.

The main definition is that of **-weighted degree **of a monomial. In particular, the -weighted degree of the monomial is . The main question is the following:

Given a degree bound , how many monomials in two variables are there that have weighted degree at most ?

In other words we need to find out how many distinct tuples with positive integers exist such that ? Below we calculate a lower bound. For notational convenience, define .

It is easy to check that the number of such tuples is

Unravelling the second sum, we get

Expanding the sum above, we obtain that the required number is

Moving the common term outside, we get

Till now we have not any approximation. However, we will do so now. Note that by the definition of , we have and . This implies, that the number of monomials of -weighted degree at most is *at least*

which is the bound that we will use in the lecture.

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