Let $f_1, \dots, f_R$ be forms of degree $k_1, \dots, k_R$ in $s$ variables. A generalised version of a conjecture by Artin states that the equations $f_1= \dots=f_R=0$ have a non-trivial $p$-adic solution for all primes $p$ provided that $s > k_1^2 + \dots + k_R^2$. This thesis proves Artin's conjecture for two diagonal forms of degree $k$ for odd primes $p$. Furthermore, it improves on this bound in the case of one diagonal cubic form and one linear form by showing that $s \ge 8$ variables are sufficient to ensure a non-trivial $p$-adic solution for all primes instead of the predicted $s \ge 11$ variables.
Keywords: $p$-adic solutions; Artin's conjecture; Diagonal forms; Additive forms; Pairs of forms; Cubic forms; Solutions in $\mathbb{Q}_p$; Analytic number theory; Diophantine equations in many variables; Congruences in many variables
