Let , and
Here is a definition of Mahler measure.
Now, let us “try” to construct a polynomial , such that .
First, consider . Hence, ; however, , so we need to calibrate the polynomial with more terms.
where and .
Let us substitute by and by ; then, we may rewrite the polynomial as
Here are the first few cases;
Obviously, for given , and share the same coefficient.
Since the polynomial is getting messier, we need a new notation; let denote the sum of product of -combination from set of . For instance, .
Now, in stead of writing down the whole family of polynomials into a table. Let us define a matrix where each entry , (the -th row and -th column) represents the coefficient of from . The entry is left blank if the coefficient is 0, just to avoid confusion.
The coefficients seem pretty random though, there is a specific pattern lurking.
Consider . Then, term in is only determined by and terms from .
For example, consider ; it is . According to the idea presented above, this term is created by
Since the degree of is all 4, let us not consider about the for now. Then,
Thus, it nicely transforms to the term, as desired.
With this nice property, I induced a following equation.
As we move back to the matrix, let us assume that has Mahler measure less than . Then, every term along the -th row is composed of integers; obviously it leads to the conclusion that for all are integers.
If Lehmer’s conjecture turns out to be true, then it implies for every , there exists no sets of that renders for every .