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Did not read the paper but it seems to me bit weird definition: that adding a constant to geometric progression makes it polynomial progression.





The abstract of the paper (https://arxiv.org/abs/1909.00309) says

> 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.


When you say "weird definition", do you mean it's a strange concept, or a strange name for that concept?

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?


I'd expect polynomial to follow definition of polynome:

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


Possibly that it doesn't seem to warrant an entirely new name? that "shifted geometric progression" or something might as well do.

See my other response to rini17 for more info.

But yeah, seems to me these "shifted geometric progressions" are actually just a special case of a more general concept of a "polynomial progression".




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