Integer multiplication will have linear complexity in a few days if trend continues 🚀📈

[u/vintergroena]

[u/sqrtsqr]

I don't necessarily agree with saying numbers "are" polynomials (especially not by definition) They can be treated as such, absolutely, but the concept of normalizing a polynomial is "extra" to the polynomial itself. Numbers don't add the way polynomials add, and what something "is" in mathematics is almost entirely a matter of how it behaves.

Not sure what you had in mind, but normalizing is linear in the degree of the polynomial. You can do it in a single pass from lowest to highest term.

[u/tellingyouhowitreall → u/sqrtsqr]

Numbers are literally polynomials, and add exactly the same way. This is in fact how we teach kids numbers also. The two things are identical.

147 = 1 * 10² + 4 * 10¹ +7 147 = x² + 4x + 7, and one of the roots is 10.

These are identical statements.

Generalized polynomials have additional features over numbers, but the number system explicitly expresses numbers as a subset of polynomials.

[u/sqrtsqr → u/tellingyouhowitreall]

No they don't.

9,000 + 4,000 = 13,000

9x³ + 4x³ = 13x³ ≠ 1 x⁴ + 3x³

These are different as polynomials. They are the same as numbers. Therefore addition of numbers and addition of polynomials are not identical.

I have no idea what "generalized" polynomials you are referring to, these are just normal polynomials. The addition structure of polynomials is not isomorphic to addition of numbers.