Did not read the paper but it seems to me bit weird definition: that adding a constant to geometric progression makes it polynomial progression.
> Let P1, … , Pm ∈ Z[y] be polynomials with distinct degrees ... progressions of the form x, x+P1(y), … , x+Pm(y)
Such progressions are a generalization of the "shifted geometric" progressions described in the article.
In particular, let P1 to Pm be y^i for i=1,...,m. Consider, say, x=0 and y = 3. Then the above gives us:
0, 3, 9, 27, ... , 3^m
Which is (almost) a geometric progression.
We can "shift" it by setting e.g. x = 1.
Though of course the polynomials can be much more general than just y^i.
The article says: > In the type of polynomial progressions studied by Peluse, you still pick a starting value, but now you add powers of another number. For example: 2, 2 + 3^1, 2 + 3^2, 2 + 3^3, 2 + 3^4. Or 2, 5, 11, 29, 83. (Her progressions also had only one term for each power, a requirement that makes them more manageable.)
I'm not entirely sure how to interpret that, since I don't see any reference to "one term for each power" in the abstract of the paper, nor in a quick glance through it.
Edit: Or do you mean that "polynomial progression" has an existing definition, and this definition is equivalent, but is a strange way of stating it? Or do you mean the definition as stated in the article is wrong / differs from the standard definition?
But yeah, seems to me these "shifted geometric progressions" are actually just a special case of a more general concept of a "polynomial progression".
x^n + c where n, c are fixed and x goes 0,1,2,... with the progression
instead of
n^x + c
..the latter I'd call exponential progression