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Remark. [ca-0005]
✍️source

This notion generalizes to commutative semirings using the approach in [izhakian2016supertropical].

References

Reference. Supertropical quadratic forms i [izhakian2016supertropical]

2016
Zur Izhakian, Manfred Knebusch, Louis Rowen
10.1016/j.jpaa.2015.05.043
https://www.sciencedirect.com/science/article/pii/S0022404915001589

@article{izhakian2016supertropical,
 title = {Supertropical quadratic forms I},
 author = {Zur Izhakian and Manfred Knebusch and Louis Rowen},
 year = {2016},
 doi = {10.1016/j.jpaa.2015.05.043},
 url = {https://www.sciencedirect.com/science/article/pii/S0022404915001589},
 journal = {Journal of Pure and Applied Algebra},
 volume = {220},
 number = {1},
 pages = {61-93},
 issn = {0022-4049}
}

Context

Forms [ca-000L]

Definition. bilinear form [ca-0003]
BilinForm
L∃∀N

Let \(R\) be a ring, \(M\) an \(R\)-module.

An bilinear form \(B\) over \(M\) is a map \(B : M \to M \to R\), satisfying:

\( B(x + y, z) = B(x, z) +B(y, z) \)
\( B(x, y + z) = B(x, y) +B(x, z) \)
\( B(a \bullet x, y) = a * B(x, y)\)
\( B(x, a \bullet y) = a * B(x, y)\)

for all \(a \in R, x, y, z \in M\).

Definition. quadratic form [jadczyk2019notes, 1.9] [ca-0004]
QuadraticForm QuadraticForm.polar
L∃∀N

Let \(R\) be a commutative ring, \(M\) a \(R\)-module.

An quadratic form \(Q\) over \(M\) is a map \(Q : M \to R\), satisfying:

\( Q(a \bullet x) = a * a * Q(x)\) for all \(a \in R, x \in M\).
there exists a companion bilinear form \(B : M \to M \to R\), such that \(Q(x + y) = Q(x) + Q(y) + B(x, y)\)

In [jadczyk2019notes], the bilinear form is denoted \(\Phi \), and called the polar form associated with the quadratic form \(Q\), or simply the polar form of \(Q\).

Remark. [ca-0005]
✍️source

This notion generalizes to commutative semirings using the approach in [izhakian2016supertropical].