An n × n sign pattern A is potentially stable (PS) if there exists a real matrix A having the sign pattern A and with all its eigenvalues having negative real parts. The identification of non-trivial necessary and sufficient conditions for potential stability remains a long standing open problem. Here we review some of the previous results and give simplified proofs for some of these results. Three techniques are given for the construction of larger order PS sign patterns from given PS sign patterns. These techniques are: construction of a sign pattern that allows a nested sequence of properly signed principal minors (a nest), bordering of a PS sign pattern with additional rows and columns, and use of a similarity transformation of a matrix that is reducible with two diagonal blocks (one of which is a stable matrix and the other a negative scalar). The minimum number of nonzero entries in an irreducible minimally PS sign pattern is determined for n = 2, . . . , 6 and for an arbitrary sign pattern that allows a nest. We also determine lower bounds for the number of nonzero entries in irreducible minimally PS sign patterns having certain sign patterns for their diagonal entries. For irreducible PS sign patterns of order at least four, a bordering construction leads to a new upper bound for the minimum number of nonzero entries.

[pt] O Modelo Padrão é uma das teorias mais bem sucedidas da física de partículas. Com a descoberta do bóson de Higgs, além de ter sido uma demonstração robusta do poder preditivo do Modelo Padrão, foi aberto um novo caminho para a investigação de nova física interagindo por meio do portal de Higgs, incluindo cenários motivados por matéria escura e bariogênese. Investigamos a estabilidade do potencial e os pólos de Landau do Modelo Padrão sob efeito da interação entre o bóson de Higgs e uma partícula escalar. Focamos no regime onde os escalares são gerados primariamente via um off-shell Higgs. Prevemos o espaço de parâmetros disponível para acessar a teoria em diferentes valores de massa do campo escalar. / [en] The Standard Model is one of the most successful theories in particle physics. With the discovery of the Higgs boson, a new pathway has been opened to investigate possible new physics interacting through the Higgs portal, including scenarios motivated by dark matter and baryogenesis. Supposing there is a neutral scalar state in the Standard Model coupled to it only through the Higgs portal, we investigate the potential stability and the Landau poles of the extended Standard Model potential. We focus on the regime in which the scalars are primarily generated via an off-shell Higgs. We predict the available parameter space to probe the theory for different mass values.