---
abstract: |
    We consider random systems of equations $x_1+ \dots  + x_k=a,$ $0 \le a \le 2$ which are interpreted as equations modulo $3.$ We show for $k \ge 15$ that the satisfiability threshold of such systems occurs where the $2-$core has density $1.$ We show a similar result for random uniquely extendible constraints over $4$ elements. Our results extend previous results of Dubois/Mandler for equations $\mod 2$ and $k=3$ and Connamacher/Molloy for uniquely extendible constraints over a domain of $4$ elements with $k=3$ arguments.

    Our proof technique is based on variance calculations, using a technique introduced Dubois/Mandler. However, several additional observations (of independent interest) are necessary.
author:
- 'Andreas Goerdt, Lutz Falke'
title: 'Satisfiability thresholds beyond $k-$XORSAT'
---

Introcuction
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Contribution
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Often constraints are equations of the type $f(x_1, \dots , x_k)=a$ where $a$ is an element of the domain considered and $f$ is a $k-$ary function on this domain, for example addition of $k$ elements. Given a formula, which is a conjunction of $m$ constraints over $n$ variables we want to find a solution. It is natural to assume that $f$ has the property: Given $k-1$ arguments we can always set the last argument such, that 