Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an ‘algeb...

Abst rac t . Let P(x) -0 (rood N) be a modular multivariate polynomial equation, in m variables, and total degree k with a small root x0. We show that there is an algorithm which determines c(~ 1) integer polynomial equations (in m variables) of total degree polynomial in cmklog N, in time polynomial in craklog N, such that each of the equations has xo as a root. This algorithm is an extension ...

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals ...

Many seminal results in Interactive Proofs (IPs) use algebraic techniques based on low-degree polynomials, the study of which is pervasive in theoretical computer science. Unfortunately, known methods for endowing such proofs with zero knowledge guarantees do not retain this rich algebraic structure. In this work, we develop algebraic techniques for obtaining zero knowledge variants of proof pr...