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.

Ways to use addGWuDivisorial:

addGWuDivisorial(List,List)

For the programmer

The object addGWuDivisorial is a method function.

The source of this document is in A1BrouwerDegrees/Documentation/UnstableGrothendieckWittClassesDoc.m2:313:0.

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

Description

References

See also

Ways to use addGWuDivisorial:

For the programmer