Researchers and practitioners of machine learning nowadays rarely overlook the potential of using explainable AI methods to understand models and their predictions. These explainable AI methods mainly focus on the importance of individual input features. However, as important as the input features themselves, are the interactions between them. Methods such as the model-agnostic but computationally expensive Friedman’s H-statistic and SHAP investigate and estimate the impact of interactions between the features. Due to computational constraints, the investigation is often limited to second-order interactions. In this paper, we present a novel, model-specific method to explain the impact of feature interactions in SVM classifiers with polynomial kernels. The method is computationally frugal and calculates the interaction importance exactly for any order of interaction. Explainability is achieved by mathematical transformation to a linear model with full fidelity to the original model. Further, we show how the model provides for both global and local explanations, and facilitates post-hoc feature selection. We demonstrate the method on two datasets; one is an artificial dataset where H-statistics requires extra care to provide useful interpretation; and one on the real-world scenario of the Wisconsin Breast Cancer dataset. Our experiments show that the method provides reasonable, easy to interpret and fast to compute explanations of the trained model. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.