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In this paper we present a deterministic algorithm for the computation of a minimal nullspace basis of an m × n input matrix of univariate polynomials over a field K with m ≤ n. This algorithm computes a minimal nullspace basis of a degree d input matrix with a cost of O ∼ (n ω md/n) field operations in K. Here the soft-O notation is Big-O with log factors removed while ω is the exponent of matrix multiplication. The same algorithm also works in the more general situation on computing a shifted minimal nullspace basis, with a given degree shift s ∈ Z n ≥0 whose entries bound the corresponding column degrees of the input matrix. In this case if ρ is the sum of the m largest entries of s, then a s-minimal right nullspace basis can be computed with a cost of O ∼ (n ω ρ/m) field operations.

The development of social media has revolutionized the way people communicate, share information and make decisions, but it also provides an ideal platform for publishing and spreading rumors. Existing rumor detection methods focus on finding clues from text content, user profiles, and propagation patterns. However, the local semantic relation and global structural information in the message propagation graph have not been well utilized by previous works.In this paper, we present a novel global-local attention network (GLAN) for rumor detection, which jointly encodes the local semantic and global structural information. We first generate a better integrated representation for each source tweet by fusing the semantic information of related retweets with the attention mechanism. Then, we model the global relationships among all source tweets, retweets, and users as a heterogeneous graph to capture the rich structural information for rumor detection. We conduct experiments on three real-world datasets, and the results demonstrate that GLAN significantly outperforms the state-ofthe-art models in both rumor detection and early detection scenarios.

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