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# addGWuDivisorial -- the divisorial sum of local degrees of a rational function

- Usage:
    `addGWuDivisorial(L1, L2)`
- Inputs:
  - L1, a list, a list of unstable Grothendieck-Witt classes representing unstable local degrees
  - L2, a list, a list of elements of the base field corresponding to the roots at which the unstable local degrees of L1 are computed
- Outputs:
  - an Unstable Grothendieck-Witt Class, the unstable Grothendieck-Witt class representing the divisorial sum of the local degrees with respect to the divisor determined by L2

## Description

Let $f/g:\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$ be a pointed rational function with zeroes $\{r_{1},\dots,r_{n}\}$ and $\{\beta_{1},\dots,\beta_{n}\}$ the unstable local $\mathbb{A}^{1}$-degrees at the $r_{i}$. The unstable global $\mathbb{A}^{1}$-degree of the rational function is not computed as the addGWu of the local unstable degrees, but as the divisorial sum [I+24].

The following example computes the divisorial sum of the rational function $\frac{x^{2}+x-2}{3x+5}$ over $\mathbb{Q}$ where the lists of unstable Grothendieck-Witt classes are given by $\{(\langle \frac{1}{3}\rangle, \frac{1}{3}), (\langle \frac{8}{3}\rangle, \frac{8}{3})\}$ and $\{-2, 1\}$.

    i1 : M1 = matrix(QQ, {{1/3}})

    o1 = | 1/3 |

                  1       1
    o1 : Matrix QQ  <-- QQ

    i2 : alpha = makeGWuClass(M1)

                   1
    o2 = (| 1/3 |, -)
                   3

    o2 : UnstableGrothendieckWittClass

    i3 : M2 = matrix(QQ, {{8/3}})

    o3 = | 8/3 |

                  1       1
    o3 : Matrix QQ  <-- QQ

    i4 : beta = makeGWuClass(M2)

                   8
    o4 = (| 8/3 |, -)
                   3

    o4 : UnstableGrothendieckWittClass

    i5 : addGWuDivisorial({alpha, beta}, {-2, 1})

    o5 = (| 1/3 0   |, 8)
          | 0   8/3 |

    o5 : UnstableGrothendieckWittClass

## References

[I+24] J. Igieobo, et. al., "Motivic configurations on the line," arXiv: 2411.15347, 2024.

## See also

- UnstableGrothendieckWittClass -- a new type intended to capture an element of the unstable Grothendieck-Witt group of a field or finite étale algebras over a field
- getGlobalUnstableA1Degree -- computes the global unstable $\mathbb{A}^{1}$-Brouwer degree of a pointed rational function $f/g:\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$
- getLocalUnstableA1Degree -- computes a local unstable $\mathbb{A}^{1}$-Brouwer degree of a pointed rational function $f/g:\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$ at a root $p\in\mathbb{P}^{1}_{k}$

## Ways to use addGWuDivisorial:

- `addGWuDivisorial(List,List)`

## For the programmer

The object addGWuDivisorial is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/UnstableGrothendieckWittClassesDoc.m2:313:0.