Add examples for all SFR gadgets

[ndattani]

For "symmetric function reduction" gadgets, the examples are almost always just copied and pasted from some previous page. Now that Tin has derived the polynomial expressions for most of these types of functions, we can give concrete examples of these quadratizations being applied.

Examples are ideally written in only 1 line, and without too much notation apart from b_i and b_{a_i}, as in this case:

This might mean we need to pick k and N small for the EKON/AKON functions. If the functions still have to be huge, we can have the degree-k function on one line, and the quadratic function on the next line.

[agu1729]

I have added examples with n=4 and k=2, which just fit within one line.

[ndattani]

Very well done!

• For consistency with other examples like Eq. 11, 25, 38, etc. it is preferable that we go from highest degree to lowest degree, so the 6b1b2b3b4 can go first and the quadratic terms can go last.
• I appreciate the brackets around S_2, S_3 and S_4 which helps to recognize how the terms can be grouped together. I'd prefer to remove the brackets around the S_2 though since it takes up less space, and is not really needed. If it didn't take up so much space I'd also prefer to remove the brackets from the S_3, but then we would need to add the coefficient "3" for each term, and it might no longer fit on one line without making the font size inconsistent.

Apart from these 2 points, I think you have made smart choices in every regard. Maybe the (N choose 2) functions can be expanded a bit so that they look more like quadratizations that people are used to seeing (not a FULL expansion, but as a product of polynomials).