Degree of the zero polynomial

[Antonio95]

Calling degree on the zero (univariate) polynomial returns 0 (here). However, the usual mathematical convention is either -infinity (with the set of degrees = N U {-infty} forming an monoid) or "undefined". In particular, setting degree(0) = 0 violates degree(fg) = degree(f) + degree(g).

Making degree return an Option would be mathematically elegant, but also programmatically cumbersome - aside from a massive breaking change. Perhaps a better solution can be found? I understand this is pretty much a non-issue, but I felt compelled to bring it up.

[emberian]

I had an annoying bug in some code expecting poly.coeffs()[..=poly.degree()] to be valid, but fortunately it was unsubtle and obvious how to fix; an extra Option would have slightly saved my effort by forcing me to remember the zero polynomial.

[Antonio95]

I agree, I have also found that being forced to treat that case separately sometimes improves the design or avoids bugs :)

[Pratyush]

Hm it is true that the degree of the zero-polynomial is indeed defined differently (I actually encountered this before when trying to do so polynomial stuff in Lean). I don't know of a good solution, however...

[Antonio95]

Something that has given me good results is for degree() to return an Option<usize>, which

• On one hand, is mathematically clean (cLEAN ;)) and forces the programmer to at least think about the zero case.
• On the other, it is not too cumbersome in practice due to how many convenience functions and syntax are built around the Option type in Rust.

But this is really just a suggestion. The breaking change would of course be sizable.

[Vincent08199]

This feels like a trade-off between mathematical correctness and simplicity.

Mathematically, the degree of the zero polynomial isn’t really 0 (it’s usually undefined or (-\infty)), so returning 0 breaks some standard properties like deg(fg) = deg(f) + deg(g).

I get why returning usize is convenient, but I think correctness should matter more here. Even if changing this to Option breaks some code, it forces users to handle the zero case explicitly, which can help avoid subtle bugs—especially in more sensitive use cases like algebra or crypto.

If a breaking change is too much, maybe adding something like is_zero() and better documentation could be a middle ground.