Stable Complete Intersections

A complete intersection of n polynomials in n indeterminates has only a finite number of zeros. In this paper we address the following question: how do the zeros change when the coefficients of the polynomials are perturbed? In the first part we show how to construct semi-algebraic sets in the parameter space over which all the complete intersection ideals share the same number of isolated real zeros. In the second part we show how to modify the complete intersection and get a new one which generates the same ideal but whose real zeros are more stable with respect to perturbations of the coefficients.Comment: 1 figur

Ideals modulo p

The main focus of this paper is on the problem of relating an ideal I in the polynomial ring Q[x_1,..., x_n] to a corresponding ideal in F_p[x_1, ..., x_n] where p is a prime number; in other words, the reduction modulo p of I. We define a new notion of sigma-good prime for I which depends on the term ordering sigma, and show that all but finitely many primes are good for all term orderings. We relate our notion of sigma-good primes to some other similar notions already in the literature. One characteristic of our approach is that enables us to detect some bad primes, a distinct advantage when using modular methods

Computing and Using Minimal Polynomials

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a “resolved problem”. But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality

Zero-dimensional families of polynomial systems

If a real world problem is modelled with a system of polynomial equations, the coefficients are in general not exact. The consequence is that small perturbations of the coefficients may lead to big changes of the solutions. In this paper we address the following question: how do the zeros change when the coefficients of the polynomials are perturbed? In the first part we show how to construct semi-algebraic sets in the parameter space over which the family of all ideals shares the number of isolated real zeros. In the second part we show how to modify the equations and get new ones which generate the same ideal, but whose real zeros are more stablewith respect to perturbations of the coefficients

Computing and Using Minimal Polynomials

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a "resolved problem". But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications

Adversarial Branch Architecture Search for Unsupervised Domain Adaptation

Unsupervised Domain Adaptation (UDA) is a key issue in visual recognition, as it allows to bridge different visual domains enabling robust performances in the real world. To date, all proposed approaches rely on human expertise to manually adapt a given UDA method (e.g. DANN) to a specific backbone architecture (e.g. ResNet). This dependency on handcrafted designs limits the applicability of a given approach in time, as old methods need to be constantly adapted to novel backbones. Existing Neural Architecture Search (NAS) approaches cannot be directly applied to mitigate this issue, as they rely on labels that are not available in the UDA setting. Furthermore, most NAS methods search for full architectures, which precludes the use of pre-trained models, essential in a vast range of UDA settings for reaching SOTA results. To the best of our knowledge, no prior work has addressed these aspects in the context of NAS for UDA. Here we tackle both aspects with an Adversarial Branch Architecture Search for UDA (ABAS): i. we address the lack of target labels by a novel data-driven ensemble approach for model selection; and ii. we search for an auxiliary adversarial branch, attached to a pre-trained backbone, which drives the domain alignment. We extensively validate ABAS to improve two modern UDA techniques, DANN and ALDA, on three standard visual recognition datasets (Office31, Office-Home and PACS). In all cases, ABAS robustly finds the adversarial branch architectures and parameters which yield best performances.Comment: Accepted at WACV 202

Upper bounds on the Laplacian spread of graphs

The Laplacian spread of a graph $G$ is defined as the difference between the largest and the second smallest eigenvalue of the Laplacian matrix of $G$. In this work, an upper bound for this graph invariant, that depends on first Zagreb index, is given. Moreover, another upper bound is obtained and expressed as a function of the nonzero coefficients of the Laplacian characteristic polynomial of a graph