Kernel methods provide an attractive framework for aggregating and learning from ranking data, and so understanding the fundamental properties of kernels over permutations is a question of broad interest. We provide a detailed analysis of the Fourier spectra of the standard Kendall and Mallows kernels, and a new class of polynomial-type kernels. We prove that the Kendall kernel has exactly two ...

Let G be a linear algebraic group over a field K of characteristic 0. An integer m is called a global degree bound for G if for every linear representation V the invariant ring K[V ] is generated by invariants of degree at most m. We prove that if G has a global degree bound, then G must be finite. The converse is well known from Noether’s degree bound.

We prove that the existential theory of any function field K of characteristic p > 0 is undecidable in the language of rings provided that the constant field does not contain the algebraic closure of a finite field. We also extend the undecidability proof for function fields of higher transcendence degree to characteristic 2 and show that the first-order theory of any function field of positive...

The randomness properties and the hierarchy of correlation coefficients are studied of approximate-maximum-length sequences, for which the characteristic polynomial is a product of several primitive polynomials. The randomness properties are almost the same as for maximum-length sequences characterized by a primitive polynomial with many terms and of the same degree. Reducible characteristic po...

For a finite cyclic Galois extension of fields K/k of degree n and a separable polynomial of degree dn or dn−1, we construct an explicit smooth compactification X → Pk of the affine normic bundle X0 given by NK/k(~z) = P (x) 6= 0, extending the map X0 → Ak, where (~z, x) 7→ x. The construction makes no assumption of the characteristic of k, making it a suitable departure point for studying the ...

Pairing based cryptography is in a dangerous position following the breakthroughs on discrete logarithms computations in finite fields of small characteristic. Remaining instances are built over finite fields of large characteristic and their security relies on the fact the embedding field of the underlying curve is relatively large. How large is debatable. The aim of our work is to sustain the...

We produce a new family of polynomials f(X) over fields k of characteristic 2 which are exceptional, in the sense that f(X)− f(Y ) has no absolutely irreducible factors in k[X,Y ] except for scalar multiples of X−Y ; when k is finite, this condition is equivalent to saying that the map α 7→ f(α) induces a bijection on an infinite algebraic extension of k. Our polynomials have degree 2e−1(2e − 1...