the q onsager algebra and the positive part of uq slˆ2 the positive part uq of uq sl 2 has a presentation by two generators x yx y that satisfy the q serre relations the q onsager algebra oqoq has a presentation by two generators a ba b that satisfy the q dolan grady relations we give two results that describe how uq and oqoq are related first we consider the filtration of oqoq whose nth component is spanned by the products of at most n generators we show that the associated graded algebra is isomorphic to uq second we introduce an algebra q q and show how it is related to both uq and oqoq the algebra q q is defined by generators and relations the generators are xi i z4 xi i z4 where z4z4 is the cyclic group of order 4 for i z4i z4 the generators xi xi 1xi xi 1 satisfy a q weyl relation and xi xi 2xi xi 2 satisfy the q serre relations we show that q q is related to uq in the following way let qeven resp qodd denote the subalgebra of q q generated by x0 x2x0 x2 resp x1 x3x1 x3 we show that i there exists an algebra isomorphism uq qeven that sends x x0x x0 and y x2y x2 ii there exists an algebra isomorphism uq qodd that sends x x1x x1 and y x3y x3 iii the multiplication map qeven qodd q u v uvu v uv is an isomorphism of vector spaces we show that q q is related to oqoq in the following way for nonzero scalars a ba b there exists an injective algebra homomorphism oq qoq q that sends a ax0 a 1x1a ax0 a 1x1 and b bx2 b 1x3b bx2 b 1x3

assessing proportionality in the proportional odds model for ordinal logistic regression the proportional odds model for ordinal logistic regression provides a useful extension of the binary logistic model to situations where the response variable takes on values in a set of ordered categories the model may be represented by a series of logistic regressions for dependent binary variables with common regression parameters reflecting the proportional odds assumption key to the valid application of the model is the assessment of the proportionality assumption an approach is described arising from comparisons of the separate correlated fits to the binary logistic models underlying the overall model based on asymptotic distributional results formal goodness of fit measures are constructed to supplement informal comparisons of the different fits a number of proposals including application of bootstrap simulation are discussed and illustrated with a data example

discrimination and classification overview publisher summary this chapter discusses discrimination and classification the problem of discrimination and classification is mainly concerned with differentiating between g g 2 mutually exclusive populations and with classifying subjects or objects on the basis of multivariate observation the emphasis of the problem is on deriving a rule that can be used to optimally classify a new subject it is an important problem in statistics and applied mathematics as its applications are found in many diverse areas in practice the distributions are rarely known and if the form of the distributions are known or assumed the parameters are seldom known thus the problem for practical applications is to develop a sample based rule for discrimination and classification the earliest and most well known method is based on the linear discriminant function due to fisher

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

decision combination in multiple classifier systems a multiple classifier system is a powerful solution to difficult pattern recognition problems involving large class sets and noisy input because it allows simultaneous use of arbitrary feature descriptors and classification procedures decisions by the classifiers can be represented as rankings of classifiers and different instances of a problem the rankings can be combined by methods that either reduce or rerank a given set of classes an intersection method and union method are proposed for class set reduction three methods based on the highest rank the borda count and logistic regression are proposed for class set reranking these methods have been tested in applications of degraded machine printed characters and works from large lexicons resulting in substantial improvement in overall correctness

the alternating pbw basis for the positive part of uq sl 2 the positive part uq of uq sl 2 has a presentation with two generators a b that satisfy the cubic q serre relations we introduce a pbw basis for uq said to be alternating each element of this pbw basis commutes with exactly one of a b qab q 1ba this gives three types of pbw basis elements the elements of each type mutually commute we interpret the alternating pbw basis in terms of a q shuffle algebra associated with affine sl2 we show how the alternating pbw basis is related to the pbw basis for uq found by damiani in 1993 the positive part uq of uq sl 2 has a presentation with two generators a b that satisfy the cubic q serre relations we introduce a pbw basis for uq said to be alternating each element of this pbw basis commutes with exactly one of a b qab q 1ba this gives three types of pbw basis elements the elements of each type mutually commute we interpret the alternating pbw basis in terms of a q shuffle algebra associated with affine sl2 we show how the alternating pbw basis is related to the pbw basis for uq found by damiani in 1993

retrospective prospective symmetry in the likelihood and bayesian analysis of case control studies summary prentice pyke 1979 established that the maximum likelihood estimate of an odds ratio in a case control study is the same as would be found by fitting a logistic regression in other words for this specific target the incorrect prospective model is inferentially equivalent to the correct retrospective model similar results have been obtained for other models and conditions have also been identified under which the corresponding bayesian property holds namely that the posterior distribution of the odds ratio is the same whether it is computed using the prospective or the retrospective likelihood in this article we demonstrate how these results follow directly from certain parameter independence properties of the models and priors and identify prior laws that support such reverse analysis for both standard and stratified designs somekeywords case controlstudy conditionalindependence hypermarkovlaw logisticregression retrospective likelihood

tree structured classification via generalized discriminant analysis abstract the problem of constructing classification rules that can be represented as decision trees is considered each object to be classified has an associated x vector containing possibly incomplete covariate information tree construction is based on the information provided in a learning sample of objects with known class identities the x vectors in the learning sample may have missing values as well procedures are proposed for each of the components of classifier construction such as split selection tree size determination treatment of missing values and ranking of variables the main idea is recursive application of linear discriminant analysis with the variables at each stage being appropriately chosen according to the data and the type of splits desired standard statistical techniques used as basic building blocks include analysis of variance linear and canonical discriminant analysis and principal component analysis a new method of tree structured classification is obtained by assem

generalized additive models some applications abstract generalized additive models have the form x fj x j where might be the regression function in a multiple regression or the logistic transformation of the posterior probability pr y 1 x in a logistic regression in fact these models generalize the whole family of generalized linear models x x where x g x is some transformation of the regression function we use the local scoring algorithm to estimate the functions fj xj nonparametrically using a scatterplot smoother as a building block we demonstrate the models in two different analyses a nonparametric analysis of covariance and a logistic regression the procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis a variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions these include confidence curves degrees of freedom estimates and approximat

the hall algebra of the category of coherent sheaves on the projective line to an abelian category a of homological dimension one satisfying certain finiteness conditions one can associate an algebra called the hall algebra kapranov studied this algebra when a is the category of coherent sheaves over a smooth projective curve defined over a finite field and observed analogies with quantum affine algebras we recover here in an elementary way his results in the case when the curve is the projective line

a goodness of fit test for the ordered stereotype model this paper presents a new goodness of fit test for an ordered stereotype model used for an ordinal response variable the proposed test is based on the well known hosmer lemeshow test and its version for the proportional odds regression model the latter test statistic is calculated from a grouping scheme assuming that the levels of the ordinal response are equally spaced which might be not true one of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories dictated by the data the proposed test takes the use of this new adjusted spacing to partition data a simulation study shows good performance of the proposed test under a variety of scenarios finally the results of the application in two examples are presented copyright 2016 john wiley sons ltd

a nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets a roadmap for a semi algebraic set s is a curve which has a non empty and connected intersection with all connected components of s hence this kind of object introduced by canny can be used to answer connectivity queries with applications for instance to motion planning but has also become of central importance in effective real algebraic geometry since it is used in higher level algorithms in this article we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets its output size and running time are polynomial in nd nlog d where d is the maximum of the degrees of the input polynomials d is the dimension of the set under consideration and n is the number of variables more precisely the running time of the algorithm is essentially subquadratic in the output size even under our assumptions it is the first roadmap algorithm with output size and running time polynomial in nd nlog d

the lusztig automorphism of the q onsager algebra abstract pascal baseilhac and stefan kolb recently introduced the lusztig automorphism l of the q onsager algebra o q in this paper we express each of l l 1 as a formal sum involving some quantum adjoints in addition i we give a computer free proof that l exists ii we establish the higher order q dolan grady relations previously conjectured by baseilhac and thao vu iii we obtain a lusztig automorphism for the current algebra a q associated with o q iv we describe what happens when a finite dimensional irreducible o q module is twisted via l

regression and ordered categorical variables read before the royal statistical society by professor r l plackett on behalf of the late professor anderson at a meeting organized by the research section on wednesday october 5th 1983 professor j b copas in the chair summary a general approach to regression modelling for ordered categorical response variables y is given which is equally applicable to ordered and unordered y the regressor variables xt xi x may be continuous or categorical the method is based on the logistic family which contains a hierarchy of regression models ranging from ordered to unordered models ordered properties of the former the stereotype model are established the choice between models is made empirically on the basis of model fit this is particularly important for assessed ordered categorical response variables where it is not obvious a priori whether or not the ordering is relevant to the regression relationship model simplification is investigated in terms of whether or not the response categories are distinguishable with respect to x the models are fitted iteratively using the method of maximum likelihood examples are given

a class of logistic regression models for multivariate binary time series abstract a logistic model for multivariate binary time series is proposed first we establish the equivalence between a log linear model for the marginal distribution of a multivariate binary random vector and logistic models for the conditional distributions of each component given the others the logistic formulation is used to describe a markov chain for each series which implies a markov model for the vector of time series a pseudolikelihood estimation procedure is presented the methods are illustrated with data on psychosomatic and psychological diagnoses for families in a health maintenance plan

hilbert functions of irreducible arithmetically gorenstein schemes abstract in this paper we compute the hilbert functions of irreducible or smooth and reduced arithmetically gorenstein schemes that are twisted anti canonical divisors on arithmetically cohen macaulay schemes we also prove some folklore results characterizing the hilbert functions of irreducible standard determinantal schemes and we use them to produce a new class of functions that occur as hilbert functions of irreducible or smooth and reduced arithmetically gorenstein schemes in any codimension

ordinal association in contingency tables some interpretive aspects abstract two families of models for ordered contingency tables goodman s association models and canonical correlation models are investigated and compared with respect to the interpretation of their parameters we show that the two families of models actually refer to different kinds of ordinal association stochastic order extremity for correlation models and stochastic order entropy for association models this difference is related to the way the two models scale interaction the scale difference is proven to be of substantial consequence especially under strong association

the maximum likelihood degree maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials we study the algebraic degree of the critical equations of this optimization problem this degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces and to the euler characteristic of the complexified complement under suitable hypotheses the maximum likelihood degree equals the top chern class of a sheaf of logarithmic differential forms exact formulae in terms of degrees and newton polytopes are given for polynomials with generic coefficients

partial separation in logistic discrimination summary the problem of maximum likelihood estimates in logistic discrimination is receiving growing attention in the literature the existence is known to be highly dependent on the data configuration observed here we focus on the practically important case where the group samples form a set of clusters that are completely separated from each other which we call partial separation the classical fisher iris data set is a good example of the problem discussed a non existence theorem is proven and a general algorithm is developed to help to distinguish separation from multicollinearity given divergence in maximum likelihood estimation

bayesian inference for the stereotype regression model application to a case control study of prostate cancer the stereotype regression model for categorical outcomes proposed by anderson j roy statist soc b 1984 46 1 30 is nested between the baseline category logits and adjacent category logits model with proportional odds structure the stereotype model is more parsimonious than the ordinary baseline category or multinomial logistic model due to a product representation of the log odds ratios in terms of a common parameter corresponding to each predictor and category specific scores the model could be used for both ordered and unordered outcomes for ordered outcomes the stereotype model allows more flexibility than the popular proportional odds model in capturing highly subjective ordinal scaling which does not result from categorization of a single latent variable but are inherently multi dimensional in nature as pointed out by greenland statist med 1994 13 1665 1677 an additional advantage of the stereotype model is that it provides unbiased and valid inference under outcome stratified sampling as in case control studies in addition for matched case control studies the stereotype model is amenable to classical conditional likelihood principle whereas there is no reduction due to sufficiency under the proportional odds model in spite of these attractive features the model has been applied less as there are issues with maximum likelihood estimation and likelihood based testing approaches due to non linearity and lack of identifiability of the parameters we present comprehensive bayesian inference and model comparison procedure for this class of models as an alternative to the classical frequentist approach we illustrate our methodology by analyzing data from the flint men s health study a case control study of prostate cancer in african american men aged 40 79 years we use clinical staging of prostate cancer in terms of tumors nodes and metastasis as the categorical response of interest

polar varieties and efficient real elimination let s 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 ldots f p this paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of s 0 for this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of s 0 this leads to a procedure which solves our algorithmic problem in time that is polynomial in the extrinsic description length of the input equations f 1 ldots f p and in a suitably introduced intrinsic geometric parameter called the degree of the real interpretation of the given equation system f 1 ldots f p

geometry of the analytic loop group abstract we introduce and study a notion of analytic loop group with a riemann hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity u g with non trivial central charge we introduce a poisson structure and study properties of its poisson dual group we prove that the hopf poisson structure is isomorphic to the semi classical limit of the center of u g it is a geometric realization of the center then the symplectic leaves and corresponding equivalence classes of central characters of u g are parameterized by certain g bundles on an elliptic curve

twisted quantum affine algebras we give a highest weight classification of the finite dimensional irreducible representations of twisted quantum affine algebras as in the untwisted case such representations are in one to one correspondence with n tuples of monic polynomials in one variable but whereas in the untwisted case n is the rank of the underlying finite dimensional complex simple lie algebra in the twisted case n is the rank of the subalgebra of fixed by the diagram automorphism the way in which such an n tuple determines a representation is also more complicated than in the untwisted case

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

algebraic degree of polynomial optimization consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials under some genericity assumptions we prove that the optimality conditions always hold on optimizers and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials we also give a general formula for the algebraic degree of the optimal coordinates the derivation of the algebraic degree is equivalent to counting the number of all complex critical points as special cases we obtain the algebraic degrees of quadratically constrained quadratic programming qcqp second order cone programming socp and p th order cone programming pocp in analogy to the algebraic degree of semidefinite programming j nie k ranestad and b sturmfels the algebraic degree of semidefinite programming math programm to appear

analysis of ordinal data in a study of endometrial cancer under a matched pairs case control design while estimating odds ratios ors in the context of dose levels of conjugated oestrogen exposure and development of endometrial cancer the categories formed by the levels of the exposure are ordinal in nature in the literature the binary logistic model is used for estimating or for each category relative to the baseline category we describe the use of two ordinal logistic models the cumulative logit and continuation ratio logit models to estimate the ors for the matched pairs case control data set of the los angeles endometrial cancer study a test for equality of the cumulative ors across the exposure levels is proposed the test statistic follows asymptotically the chi square distribution

bootstrap choice of estimators in parametric and semiparametric families an extension of eic ishiguro sakamoto et kitagawa 1997 ont propose le eic comme une extension du critere d akaike aic le eic est construit en corrigeant le biais de la log vraisemblance consideree comme un estimateur de l information de kullback leibler par la technique du bootstrap nous developpons ce critere dans un cadre d estimation semi parametrique multivarie et nous montrons qu il peut etre utilise pour choisir un estimateur parmi une famille d estimateurs parametriques et semi parametriques une etude de simulation basee sur un modele de regression montre que l eic est meilleur que les criteres usuels bien que la validation croisee appliquee a la vraisemblance donne des resulats tres proches sauf dans le cas d echantillons de faible taille nous avons applique le eic afin d estimer l evolution moyenne de la charge virale d enfants infectes par le vih

loop realizations of quantum affine algebras we give a simplified description of quantum affine algebras in their loop presentation this description is related to drinfeld s new realization via halves of vertex operators we also define an idempotent version of the quantum affine algebra which is suitable for categorification

a note on regression analysis of ordinal data with variability of classification summary the effect of observation variability in the classification of an ordinal response is examined within the framework of the proportional hazards regression model for ordinal data a model is proposed which incorporates this type of variability and the behaviour of the proportional hazards model when this new model is appropriate is examined four data sets are used to illustrate the practical application of the model

classification efficiency of multinomial logistic regression relative to ordinal logistic regression abstract classification procedures are useful for the prediction of a response or outcome as a result of knowledge of the levels of one or more independent or predictor variables the procedure is said to classify the possibly multivariate observation to a level of the response variable an example might be the prediction of whether an individual will be well suffer a nonfatal heart attack or suffer a fatal heart attack this prediction might be made on the basis of the levels of various independent variables such as weight blood pressure and serum cholesterol to name a few the three response categories of the aforementioned example are ordinal an example of three nonordered response categories might be as follows well death from heart attack and death from cancer there is some recent interest in ordinal classification procedures it is reasonable to assume that when the response variable is ordinal inclusion of ordinality in the classification model to be estimated should improve mode

penalized maximum likelihood estimation in logistic regression and discrimination summary maximum likelihood estimation of the parameters of the binary logistic regression model for pr h i x is discussed with separate discussion of sampling from i the conditional distribution of h given x ii the joint distribution of h and x and iii the conditional distribution of x given h difficulties associated with continuous x in the latter sampling scheme are discussed to avoid these penalized maximum likelihood estimates are introduced which give estimates of the logistic parameters and a nonparametric spline estimate of the marginal distribution of x extensions to multinomial logistic regression are outlined

prediction of ordinal outcomes when the association between predictors and outcome differs between outcome levels there are a number of regression models which are widely used to predict ordinal outcomes the commonly used models assume that all predictor variables have a similar effect at all levels of the outcome variable if this is not the case for example if some variables predict susceptibility to a disease and others predict the severity of the disease then a more complex model is required one possibility is the multinomial logistic regression model which assumes that the predictor variables have different effects at all levels of the outcome variable an alternative is to use the stereotype family of regression models a one dimensional stereotype model makes the assumption that the effect of each predictor is the same at all outcome levels however it is possible to fit stereotype models with more than one dimension up to a maximum of min k 1 p where k is the number of outcome categories and p is the number of predictor variables a stereotype model of this maximum dimension is equivalent to a multinomial logistic regression model in that it will produce the same predicted values and log likelihood if there are sufficient outcome levels and or predictor variables there may be a number of stereotype models of differing dimension the method is illustrated with an example of prediction of damage to joints in rheumatoid arthritis

reduction from cost sensitive ordinal ranking to weighted binary classification we present a reduction framework from ordinal ranking to binary classification the framework consists of three steps extracting extended examples from the original examples learning a binary classifier on the extended examples with any binary classification algorithm and constructing a ranker from the binary classifier based on the framework we show that a weighted 0 1 loss of the binary classifier upper bounds the mislabeling cost of the ranker both error wise and regret wise our framework allows not only the design of good ordinal ranking algorithms based on well tuned binary classification approaches but also the derivation of new generalization bounds for ordinal ranking from known bounds for binary classification in addition our framework unifies many existing ordinal ranking algorithms such as perceptron ranking and support vector ordinal regression when compared empirically on benchmark data sets some of our newly designed algorithms enjoy advantages in terms of both training speed and generalization performance over existing algorithms in addition the newly designed algorithms lead to better cost sensitive ordinal ranking performance as well as improved listwise ranking performance

bayesian model comparison for the order restricted rc association model association models constitute an attractive alternative to the usual log linear models for modeling the dependence between classification variables they impose special structure on the underlying association by assigning scores on the levels of each classification variable which can be fixed or parametric under the general row column rc association model both row and column scores are unknown parameters without any restriction concerning their ordinality however when the classification variables are ordinal order restrictions on the scores arise naturally under such restrictions we adopt an alternative parameterization and draw inferences about the equality of adjacent scores using the bayesian approach to achieve that we have constructed a reversible jump markov chain monte carlo algorithm for moving across models of different dimension and estimate accurately the posterior model probabilities which can be used either for model comparison or for model averaging the proposed methodology is evaluated through a simulation study and illustrated using actual datasets

properties of ideal point classification models for bivariate binary data the ideal point classification ipc model was originally proposed for analysing multinomial data in the presence of predictors in this paper we studied properties of the ipc model for analysing bivariate binary data with a specific focus on three quantities 1 the marginal probabilities 2 the association structure between the two binary responses and 3 the joint probabilities we found that the ipc model with a specific class point configuration represents either the marginal probabilities or the association structure however the ipc model is not able to represent both quantities at the same time we then derived a new parametrization of the model the bivariate ipc bipc model which is able to represent both the marginal probabilities and the association structure like the standard ipc model the results of the bipc model can be displayed in a biplot from which the effects of predictors on the binary responses and on their association can be read we will illustrate our findings with a psychological example relating personality traits to depression and anxiety disorders

multiple group logistic discrimination among the many possible approaches suggested for statistical discrimination the logistic method can be classified midway between fully distributional solutions of which the assumption of multivariate normality is a classical example 16l and the distribution free techniques using for instance kernel or nearest neighbor methods l 32 therefore it is often called a partially parametric or partially distributional method l l this central position of the logistic model makes it one of the most attractive and widely used tools for solving regression and discrimination problems indeed since there are fewer distributional assumptions than for fully parametric models the logistic method is applicable to a larger family of multivariate distributions involving both discrete and continuous variables moreover in spite of its wide applicability and generality the method remains feasible and easy to use in contrast with nonparametric methods this paper is intended to review the basic ideas and principles of logistic discrimination 415 22 25 and also to bring additional results to some of the queries raised by professor anderson before his death we restrict our attention to discrimination between qualitatively distinct groups and do not envisage the case where groups are quantitatively distinct or ordered 3 12 14 we dedicate this paper to our friend and mentor the late professor j a anderson for his fundamental contribution to discriminant analysis and for his continuous support of our research efforts

a new model for ordinal pain data from a pharmaceutical study for the pain data analysed previously by cox and chuang this paper proposes a new model that assumes monotone scores for ordered response categories this proposed model possesses several attractive features and allows a stochastic ordering of the drugs under comparison such a model also provides insight regarding the ordinal scale used to classify response estimation of the parameters in the model is obtained by use of bmdp3r

are ordinal models useful for classification there is recent interest in classification procedures intended for use only when the response is ordinal ordinal response however is evident in the parameters estimated by either multinomial logistic or normal discriminant analyses both of which classify either ordinal or non ordinal responses further there may be harm in applying ordinal models inappropriately and ample opportunity to assume mistakenly ordinality in real data sets therefore it becomes important to ascertain whether there is benefit obtained in the appropriate application of ordinal models this paper presents the results of a simulation study designed to compare classification accuracy of various models we show that ordinal models classify less accurately than the multinomial logistic and normal discriminant procedures under a variety of circumstances until further studies become available we presently conclude that ordinal models confer no advantage when the main purpose of the analysis is classification

discriminant analysis when all variables are ordered determination of the equation that relates an ordered dependent variable to ordered independent variables is sought one solution non parametric discriminant analysis npd involves obtaining the best monotonic step function by means of a computer search procedure although one can use alternative selection criteria in obtaining the equation the illustrative examples use absolute distance this paper compares the prediction procedures obtained from npd with those from linear discriminant analysis linear regression with and without transformed variables and logistic regression we show that npd is analogous to regression tree analysis with incorporation of ordered variables and monotonicity we use various prediction functions to predict the example data the data using the leave one out technique and a verification set consistently non parametric discriminant analysis performs as good as or better than the tested alternatives

on the complexity of computing critical points with grobner bases computing the critical points of a polynomial function q in mathbb q x 1 ldots x n restricted to the vanishing locus v subset mathbb r n of polynomials f 1 ldots f p in mathbb q x 1 ldots x n is of first importance in several applications in optimization and in real algebraic geometry these points are solutions of a highly structured system of multivariate polynomial equations involving maximal minors of a jacobian matrix we investigate the complexity of solving this problem by using grobner basis algorithms under genericity assumptions on the coefficients of the input polynomials the main results refine known complexity bounds which depend on the maximum d max deg f 1 ldots deg f p deg q to bounds which depend on the list deg f 1 ldots deg f p deg q we prove that the grobner basis computation can be performed in delta o log a log g arithmetic operations in mathbb q where delta is the algebraic degree of the ideal vanishing on the critical poin