Modular Invariance of Strings on PP-Waves with RR-flux

Selection of Scale-Invariant Parts for Object Class RecognitionGy.Dork´o and C.SchmidINRIA Rhˆo ne-Alpes,GRA VIR-CNRS655,av.de l’Europe,38330Montbonnot,Francedorko,schmid@inrialpes.frAbstractThis paper introduces a novel method for constructing and selecting scale-invariant object parts.Scale-invariant local descriptors arefirst grouped into basic parts.A clas-sifier is then learned for each of these parts,and feature selection is used to determine the most discriminative ones. This approach allows robust part detection,and it is invari-ant under scale changes—that is,neither the training im-ages nor the test images have to be normalized.The proposed method is evaluated in car detection tasks with significant variations in viewing conditions,and promising results are demonstrated.Different local re-gions,classifiers and feature selection methods are quanti-tatively compared.Our evaluation shows that local invari-ant descriptors are an appropriate representation for object classes such as cars,and it underlines the importance of feature selection.1.IntroductionRecognizing general object classes and parts is one of the most challenging problems in computer vision.The combi-nation of computer vision and machine learning techniques has recently led to significant progress[1,17,18],but exist-ing approaches are based onfixed-size windows and do not make use of recent advances in scale-invariant local feature extraction[6,8].Thus,they require normalizing the train-ing and test images.We propose in this paper a method for selecting discrimi-native scale-invariant object parts.Figure1(a)demonstrates the importance of feature selection.It shows the output of a scale-invariant operator forfinding significant circu-lar patches in images[6].In this context,it is natural to define object parts in terms of clusters of patches with sim-ilar brightness patterns.However,consider the two patchesmarked in black in thefigure.The corresponding patterns are very close,but one of the patches lies on a car,while the other lies in the background.This shows that the cor-responding part is not discriminative for cars(in this en-vironment at least).To demonstrate the effect of the pro-posed feature selection method,Fig.1(b)shows the initially detected features(white)and discriminative descriptors de-termined by feature selection(black).These are the ones which should be used in afinal,robust detectionsystem.(a)(b)Figure1.Illustration of feature selection(seetext for details).1.1.Related WorkMost appearance-based approaches to object class recognition characterize the object by its whole image [9,15].They are not robust to occlusion and also suffer 1from a lack of invariance.Furthermore,these methods are only applicable to rigid objects and either they require pre-liminary segmentation,or windows have to be extracted for different locations,scales and rotations.The representa-tion is also high-dimensional,therefore many learning tech-niques cannot be used.To overcome these problems the use of local features is becoming increasingly popular for object detection and recognition.Weber et al.[18]use localized image patches and ex-plicitly compute their joint spatial probability distribution. This approach has recently been extended to include scale-invariant image regions[11].Agarwal and Roth[1]first learn a vocabulary of parts,determine spatial relations on these parts and use them to train a Sparse Network of Win-nows(SNoW)learning zebnik et al.[5]take advantage of local affine invariants to represent textures.Some recent methods combine feature selection and lo-cal features.Viola and Jones[17]select rectangular(Haar-like)features with an AdaBoost trained classifier.Chen et al.[3]also use this boosting approach for components learned by local non-negative matrix factorization.Amit and Geman[2]employ small,localized and oriented edges and combine them with decision trees.Mahamud and Hebert[7]select discriminative object parts and develop an optimal distance measure for nearest neighbor search. Rikert et al.[12]use a mixture model,but only keep the discriminative clusters and Schmid[14]selects significant texture descriptors in a weakly supervised framework.Both approaches select features based on their likelihood.Ull-mann et al.[16]use image fragments and combine them with a linear discriminative type classification rule.Their selection algorithm is based on mutual information.1.2.OverviewThefirst step of our approach is the detection of scale-invariant interest points(regions)and the computation ofa rotation-invariant descriptor for each region(cf.section2.1).These descriptors are then clustered and we obtaina set of parts each of which is described by a classifier (cf.section2.2).Finally,we select a set of discriminative parts/classifiers(cf.section3).An experimental evaluation compares different approaches to region extraction,classi-fication and selection(cf.Section4).Finally in Section5 we conclude and outline our future work.2.Object-Part ClassifiersIn the following wefirst describe how to compute in-variant descriptors and then explain how to learn object part classifiers.2.1.Scale-Invariant DescriptorsTo obtain invariant descriptors we detect scale-invariant interest points(regions)and characterize each of them by a scale,rotation and illumination invariant descriptor.Scale-invariant detectors.We have used two differ-ent scale-invariant detectors:Harris-Laplace[8]and DoG (Difference-of-Gaussian)[6].Harris-Laplace detects multi-scale Harris points and then selects characteristic points in scale-space with the Laplace operator.DoG interest points[6]are local scale-space maxima of the Difference-of-Gaussian.The image locations(regions)selected by the two detec-tors differ:The DoG detectorfinds blob-like structures and Harris-Laplace detects corners and highly textured points. Examples for detection are shown in thefirst column of Fig-ure7.Scale and rotation invariant descriptors.The output of the two detectors are scale-invariant regions of different sizes.These regions arefirst mapped to circular regions of afixed-sized radius.Point neighborhoods which are larger than the normalized region,are smoothed before the size normalization.Rotation-invariance is obtained by rotation in the direction of the average gradient orientation(within a small point neighborhood).Affine illumination changes of the pixel intensities are eliminated by normal-ization of the image region with the mean and the standard deviation of the intensities within the point neighborhood. These normalized regions are then described by the SIFT descriptor(Scale Invariant Feature Transform)[6].SIFT is computed for8orientation planes and each gradient image is sampled over a4x4grid of locations.The resulting de-scriptor is of dimension128.2.2.ClassifiersObject-part classifiers are learned from sets of similar descriptors.These sets are obtained automatically by clus-tering local invariant descriptors.Figure2shows a few im-ages of two different clusters.The top row displays a“tire”cluster and the bottom row a“front window”cluster.We have used two types of classifiers:Support Vector Machines(SVMs)and classification based on a Gaussian mixture model(GMM).The training data consists of posi-tive and negative descriptors.Note that the descriptors are labeled manually.Support Vector Machine.Each object part is described by a separate SVM.A descriptor is classified as a part,if the the corresponding SVM has a positive response.The SVMs are trained as follows.Thefirst step is to de-termine groups of similar descriptors.We cluster the pos-itive training descriptors with a hierarchical clustering al-gorithm.The number of clusters is set to300.We thenFigure2.A few images of two different clusters.Thefirst row shows a cluster which represents “tires”.The second row shows a cluster which contains regions detected in the“front window”.learn a linear SVM[4]for each positive cluster.The SVM is trained with all descriptors of the positive cluster and a subset of the negative descriptors.This subset are the medi-ans of negative clusters.Note that this pre-selection of the negative samples is necessary.Otherwise we would have highly unbalanced training sets,which can not be handled by current state of the art SVM techniques.Gaussian mixture model.The distribution of the train-ing descriptors is described by a Gaussian mixture model.Each Gaussian corresponds to an “object-part”.A descriptor is assigned to the most likely Gaussian,i.e.it is classified as the corresponding part. Each is assumed to be a Gaussian with mean and covariance matrix.We use the EM algorithm to estimate the parameters of the mixture model,namely the means,covariances,and mixing weights.EM is initialized with the output of the-means algorithm.In this work,we use the600components to describe the train-ing set which includes positive and negative descriptors.We use all positive and randomly choose the same number of negative descriptors.We limit the number of free parame-ters in the optimization by using diagonal Gaussians.This restriction also helps prevent the covariance matrices from becoming singular.3.Feature SelectionGiven a set of classifiers,we want to rank them by their distinctiveness.Here,we use two different feature selection techniques:likelihood ratio and mutual information.These techniques assign a score to each classifier depending on its performance on a validation set.The two feature selection methods are based on the prob-abilities described in the following.Let be a classifier and the object to be recognized(detected).is the probability that classifies a object descriptor correctly(i.e the true positives for over the number of positives descriptors).is the probability of non-objects descriptors being accepted by classifier.Likelihood ratio.A classifier is representative of an ob-ject class if it is likely to be found in the class,but unlikely to be detected in non-class images.The likelihood ratio of classifier is defined by:Mutual information.Mutual information[10]selects in-formative features.Mutual information between the classifier and the object class is defined by:For both feature selection methods presented above,the higher the score the more relevant it is.The difference be-tween the two methods is illustrated by Figure3.The image is one of the test images and regions are detected with the DoG detector.The top4rows show the descriptors classified as object parts by the best SVM classifiers.We can see that the likelihood selects very specific features which are highly discriminative.For example there is no car feature detectedLikelihood Mutual Information1selected part,SVMno regions12correct+1incorrect5selected parts,SVM1correct16correct,6incorrect10selected parts,SVM2correct21correct,7incorrect25selected parts,SVM25correct,7incorrect74correct,33incorrect50selected parts,GMM7correct,1incorrect56correct,38incorrect100selected parts,GMM19correct,8incorrect86correct,83incorrect Figure parison of feature selection with likelihood ratio and mutual information.by the“best”classifier in the case of likelihood ratio.This feature is very specific and only detected on certain cars. In contrast mutual information selects informative features. For example thefirst selected features already classifies13 descriptors as object descriptors.Note that one of them is incorrect.If we look at the overall performance of the two feature selection methods,we can observe that the likeli-hood ratio performs slightly better than mutual information. Note however that fewer classifiers are used in the case of mutual information.This is confirmed by the images in the 2bottom rows which show the results for the best GMM classifiers as well as by the quantitative evaluation in Sec-tion4.Note that to obtain similar results for GMM we have to use more classifiers.This is due to the fact that there are twice as many classifiers and that they are in general more specific.4.ExperimentsIn the previous sections we have presented several tech-niques for the different steps of our approach.We now eval-uate these techniques in the context of car detection.We then present car detection results for a few test images.4.1.Set-upOur training database contains images of cars with a relatively large amount of background(more that on average).We have marked the cars in these images man-ually.Note that the car images can be at different scale levels and do not require normalization.We extract scale-invariant interest points(regions)with the DoG detector and Harris-Laplace.For DoG we obtained positive and negative regions.For Harris-Laplace we detectedpositive and negative regions.The test images were taken independently and contain unseen cars and unseen background.We have used im-ages which often contain several cars and a large amount of background.To evaluate and compare different meth-ods,we marked them manually.We therefore know that the test images contain positive and negative de-scriptor if the DoG detector is used and and descriptors for Harris-Laplace.4.2.Evaluation of Different MethodsIn the following we evaluate our approach and compare the performance of different techniques.The comparison criterion is true positive rate(the number of positive de-scriptors retrieved over the total number of positive descrip-tors)against false positive rate(the number of false positives over the total number of negatives descriptors).Classification and Feature selection.Figure4comparesthe performance of two different classification techniques and two different feature selection criteria.Regions are ex-tracted with the DoG detector.Fig.4shows the ROC curve (true positive rate against false positive rate).We can ob-serve that the combination of Gaussian mixture model and likelihood ratio performs best.The second best is the curve for SVM and likelihood ratio which performs slightly better than SVM and mutual information.Results for the combi-nation of mixture model and mutual information are signif-icantly worse.This can be explained by the fact that the classifiers are mostly specific.Fig.5(a)and (b)compare the criteria true positive rate and false negative rate separately as a function of the number of selected classifiers.As ex-pected mutual information has a higher true positive rate and the false negatives rate is better (lower)for the likeli-hood ratio.0.20.40.60.8100.20.40.60.81T r u e P o s i t i v e R a t eFalse Positive RateDoG, SVM, Likelihood DoG, SVM, Mutual Inf.DoG, GMM, Likelihood DoG, GMM, Mutual Inf.Figure parison of the performance of the likelihood ratio and the mutual informa-tion for SVM and GMM.Regions are extracted with the DoG detector.Descriptors.We have also compared the performance of the two detectors DoG and Harris-Laplace.Figure 6shows the results for Harris-Laplace.We can observe that the rank-ing of the different combinations of classifier and feature selection techniques are the same as for DoG.Furthermore,Harris-Laplace and DoG show a similar performance.How-ever,we have noticed that the behavior depends on the test image.Furthermore,Harris-Laplace detects less points on the background and therefore detects more true positives than DoG for a fixed number of false positives.4.3.Car Recognition/DetectionIn this section we illustrate the performance of our ap-proach with two examples.Figure 7shows results for DoG and Harris-Laplace as well as the two classification tech-niques.The first column displays the detected regions.The0.2 0.4 0.6 0.8 1 050100150200250300T r u e P o s i t i v e R a t ePartsDoG, SVM, Likelihood DoG, SVM, Mutual Inf.a0.2 0.4 0.6 0.8 1 050100150 200250300F a l s e P o s i t i v e R a t ePartsDoG, SVM, Likelihood DoG, SVM, Mutual Inf.bFigure parison of the performance of the likelihood ratio and the mutual informa-tion for SVM.Regions are extracted with the DoG detector.0.20.40.60.810.20.40.60.81T r u e P o s i t i v e R a t eFalse Positive RateHarris, SVM, Likelihood Harris, SVM, Mutual Inf.Harris, GMM, Likelihood Harris, GMM, Mutual Inf.Figure parison of the performance of the likelihood ratio and the mutual informa-tion for SVM and GMM.Regions are extracted with the Harris-Laplace detector.second column shows the results of the25best parts ob-tained with SVM and selected by the likelihood ratio.The third column displays the results of the100best parts for GMM and likelihood ratio.We can see that the method al-lows to select car features.It can be further improved by adding relaxation.Relaxation.The descriptors selected in Figure7are sparsely distributed over the object(car).We would like to obtain a dense feature map which permits segmentation of the object.Given the selected features,we can use the order of se-lection to assign a probability to each descriptor.A de-scriptor which is classified by a more discriminative feature is assigned a higher probability.We can then use relax-ation[13]to improve the classification of the descriptors. Relaxation reinforces or weakens the probabilities depend-ing on the probabilities of the nearest neighbors(5in our experiments).Figure8shows the descriptors classified as car features after applying the relaxation algorithm.Initial results based only on feature selection are shown in Figure7 (DoG,SVM and likelihood).Compared to these initial re-sults,we can clearly observe that more features are detected on the cars and less on the background,that is the overall performance is significantly improved.Further improve-ment is possible by integrating spatial constraints into the neighborhood relations of the relaxation process.5.Conclusion and Future workIn this paper,we have introduced a method for construct-ing object-part classifiers and selecting the most discrimi-nant ones.Object-parts are invariant to scale and rotation as well as illumination changes.Alignment of the training and test images is therefore not necessary.This paper has also illustrated the importance of feature selection and has compared different techniques.This comparison shows that likelihood is well suited for object recognition and mutual information for focus of attention mechanisms,that is rapid localization based on a few classifiers.Learning of the parts is unsupervised,but the descrip-tors are manually marked as positive and negative.We plan to extend the approach to the weakly supervised case where the descriptors are unlabeled and only the images are marked as positive or negative.This should be straightfor-ward in the case of classification with a Gaussian mixture model.AcknowledgmentsThis work was supported by the European project LA V A. We thank S.Agarwal for providing the carimages.Figure8.Improved results for object detec-tion by adding relaxation.References[1]S.Agarwal and D.Roth.Learning a sparse representationfor object detection.In ECCV,2002.[2]Y.Amit and D.Geman.A computational model for visualselection.Neural Computation,11(7):1691–1715,1999. [3]X.Chen,L.Gu,S.Li,and H.-J.Zhang.Learning represen-tative local features for face detection.In CVPR,2001. [4]N.Cristianini and J.Shawe-Taylor.Support Vector Ma-chines.Cambridge University Press,2000.[5]zebnik,C.Schmid,and J.Ponce.Sparse texture rep-resentation using affine-invariant neighborhoods.In CVPR, 2003.[6] D.G.Lowe.Object recognition from local scale-invariantfeatures.In ICCV,pages1150–1157,1999.[7]S.Mahamud and M.Hebert.The optimal distance measurefor object detection.In CVPR,2003.[8]K.Mikolajczyk and C.Schmid.Indexing based on scaleinvariant interest points.In ICCV,pages525–531,2001. [9] C.Papageorgiou and T.Poggio.A trainable system for ob-ject detection.IJCV,38(1):15–33,2000.[10] A.Papoulis.Probability,Random Variables,and StochasticProcesses.McGraw Hill,1991.[11]R.Fergus,P.Perona and A.Zisserman.Object class recog-nition by unsupervised scale-invariant learning.In CVPR, 2003.[12]T.Rikert,M.Jones,and P.Viola.A cluster-based statisti-cal model for object detection.In ICCV,pages1046–1053, 1999.[13] A.Rosenfeld,R.Hummel,and S.Zucker.Scene labeling byrelaxation operations.IEEE Transactions on Systems,Man and Cybernetics,6:420–433,1976.[14] C.Schmid.Constructing models for content-based imageretrieval.In CVPR,2001.[15]K.Sung and T.Poggio.Example-based learning for view-based human face detection.IEEE Transactions on PAMI, 20(1):39–51,1998.[16]S.Ullman,E.Sali,and M.Vidal-Naquet.A fragment-basedapproach to object representation and classification.In4th International Workshop on Visual Form,Capri,Italy,2001.[17]P.Viola and M.Jones.Rapid object detection using aboosted cascade of simple features.In CVPR,volume I, pages511–518,2001.[18]M.Weber,M.Welling,and P.Perona.Unsupervised learn-ing of models for recognition.In ECCV,pages18–32,2000.Detection25parts,SVM,Likelihood100parts,GMM,LikelihoodDoG508regions32correct,14incorrect18correct,8incorrectHarris-Laplace364regions49correct,37incorrect15correct,11incorrectDoG431regions30correct,18incorrect13correct,17incorrectHarris-Laplace277regions43correct,36incorrect32correct,13incorrect Figure7.Results for two test images.The left column shows the interest regions detected with DoG and Harris-Laplace.The middle column displays the descriptors classified by the25best SVM classifiers selected with the likelihood ratio.The right column shows the results for the100best GMM classifiers selected with likelihood ratio.。

‘中国科技史杂志“第45卷(2024年)第1期:67 76The Chinese Journal for the History of Science and Technology ㊀Vol.45(2024)No.1伽罗瓦定理真的错了吗?杨保强(延安大学数学与计算机科学学院,延安716000)摘㊀要㊀1830年,伽罗瓦提出有关本原方程的一个定理㊂在数学史上,很多学者认为该定理对于本原群的刻画是错误的,但有一些研究者猜想伽罗瓦的定理可能无误,也许伽罗瓦的本原群隐含地假设了二重传递性㊂本文通过引入与之相应的 二重传递方程 的概念,利用古证复原或数学实操的方法复原伽罗瓦对于本原方程的真实认识,证实伽罗瓦定理中的 本原方程 实际所指乃二重传递方程,对应于二重传递群的结果,伽罗瓦的定理并无差错㊂关键词㊀伽罗瓦㊀本原方程㊀二重传递群㊀数学实操㊀古证复原中图分类号㊀N09ʒO151.1文献标识码㊀A㊀㊀㊀㊀文章编号㊀1673-1441(2024)01-0067-10㊀㊀㊀收稿日期:2023-01-25;修回日期:2023-07-03㊀㊀㊀作者简介:杨保强,1993年生,延安大学数学与计算机科学学院讲师,研究方向为近现代数学史㊂㊀㊀㊀基金项目:延安大学博士科研启动项目 本原方程的历史研究 (项目编号:YDBK2022-64);国家自然科学基金地区科学基金项目 非欧几何学的若干历史问题研究 (项目编号:12161086)㊂1　问题的提出1828年,挪威数学家阿贝尔(Niels Henrik Abel,1802 1829)提出:代数方程的基本问题是根式可解方程的确定和分类([1],pp218 219)㊂阿贝尔之后,伽罗瓦(Évariste Galois,1811 1832)㊁若尔当(Camille Jordan,1838 1922)等很多数学家都是在这个问题的驱动下去研究代数方程的㊂([2],pp34 38)在彻底解决了素数次不可约方程根式可解的问题之后,伽罗瓦将寻找合数次根式可解的不可约方程的问题简化为寻找素数幂次根式可解的本原方程的问题([3],p286)㊂其中,本原方程(primitive equations)是指方程的伽罗瓦群为本原群的一类不可约方程([4],p119),它并非现代意义中的本原多项式方程㊂1830年4月,伽罗瓦发表了他研究本原方程的结果㊂为刻画素数幂次根式可解的本原方程的类型,伽罗瓦给出这样一个定理:伽罗瓦定理.除了9次和25次方程,素数幂次本原方程根式可解的必要条件是,已知它的两个根,其余的根都可以表示为它们的有理函数㊂([5],p271)伽罗瓦对他的定理并没有太多的解释,也未留下证明㊂两个月后,伽罗瓦更加精确地86中㊀国㊀科㊀技㊀史㊀杂㊀志45卷表述了这一必要条件,并认定它也是该定理的一个充分条件㊂([6],p435)伽罗瓦是通过对素数幂次根式可解的本原方程的伽罗瓦群的刻画来表述它的可解条件的㊂上升到现代群论的认识,他的定理等价于:除了9次和25次方程,该方程的伽罗瓦群同构于1维仿射半线性群AΓL(1,F p n)的一个子群([7],p35)㊂然而,事实并非如此㊂正确的结果应当是,它的伽罗瓦群同构于n维仿射一般线性群AGL(n,F p)的一个子群([8],p452)㊂当n 不等于1时,这两个群是完全不同的㊂因此,正如英国学者纽曼(Peter M.Neumann, 1940 2020)所述,伽罗瓦的结果 是完全错误的 ([7],p35)㊂事实上,伽罗瓦定理的错误最早由法国数学家若尔当在1867年指出㊂([9], p108)若尔当发现,素数平方次的本原方程并不适用伽罗瓦的定理,以此作为伽罗瓦定理的反例,揭示了它的错误性: 伽罗瓦已经提出,根式可解的本原方程属于一种类型,但不包括9次与25次方程㊂根据上面的论述,我们必须取几乎完全相反的断言 ([10], p113)㊂若尔当之后,了解这段历史的研究者纽曼也对伽罗瓦的定理持否定的态度㊂([11],p49)尽管如此,一些学者猜想伽罗瓦的定理也许并无差错㊂1957年,群论专家于佩尔(Bertram Huppert,1927 )得到关于可解二重传递置换群的一个分类定理:于佩尔定理.除了当p n是32,52,72,112,232,34时的情形,任何p n次[p为素数]的可解二重传递置换群同构于1维仿射半线性群AΓL(1,F p n)的一个子群㊂([12], p379)本原群未必都是二重传递的,但是,如果伽罗瓦的 本原 (primitive)实际所指乃 二重传递 (doubly transitive)的话,那么,不考虑伽罗瓦未找到的例外,伽罗瓦的定理已经近乎于佩尔的结果了,而不至于说它存在根本性的错误㊂就像考克斯(David A.Cox, 1948 )2012年评述的那样:若尔当的结果揭示出伽罗瓦对可解本原群的刻画有一定的差距㊂然而,伽罗瓦或许隐含地假设了二重传递性㊂如果是这样的话,那么他的描述(除了上面的例外)就非常接近于完整了㊂([13],p57)在2002年的文章中,拉德洛夫(Ivo Radloff)也假定伽罗瓦的定理是作了二重传递性的假设,只有在此条件下,伽罗瓦的定理才不致有误㊂([14],p133)根据于佩尔的定理,伽罗瓦的定理在二重传递群的条件下才能成立,由这两个数学定理的相似性,拉德洛夫和考克斯猜测伽罗瓦的本原群或许假设了二重传递性㊂如此猜想有一定的合理性,因为,伽罗瓦将 本原群 理解为二重传递群这一点是可能的㊂原因在于,在群论研究的初期,伽罗瓦对于本原群的认识本来就很模糊,比如,他也曾将本原群混同于 拟本原群 (其任意非平凡的正规子群皆为传递群)的概念[15];而且,在生命的最后,伽罗瓦给出了关于可解本原群的正确的定理([11],p87),伽罗瓦的 本原群 概念可能发生了改变,伽罗瓦之前的定理很大可能是他将 本原群 限制于二重传递群得到的㊂不过,归于历史问题的分析,想要证实伽罗瓦的本原群作了二重传递性的假设,无疑是困难的:一方面,伽罗瓦并未证明他的定理,除了定理本身之外, 很难确切地知道伽罗瓦是怎么想的 ([8],p452);另一方面,伽罗瓦没有对他的本原群的概念作出任何解释,㊀1期杨保强:伽罗瓦定理真的错了吗?96更何况二重传递群的概念在伽罗瓦逝世三十年后才出现㊂那么,如何证实伽罗瓦的定理是没错的猜想呢?前人是将伽罗瓦的定理上升至现代群论的认识而猜想它可能没错的,但伽罗瓦最初的定理针对的却是本原方程㊂因此,无论如何,我们都必须返回伽罗瓦的原始表述,通过探析伽罗瓦定理当中的本原方程概念来厘定伽罗瓦定理现存的争议㊂伽罗瓦到底错了还是没有?这个问题在很大程度上取决于伽罗瓦的定理或许没错的猜想能否得到证实㊂如果不能证实这一点,伽罗瓦可能没错的猜想将依旧止步于猜想,数学史界对于天才数学家伽罗瓦的评定也将存在两种截然相反且并不容中的观点,这是史学研究的科学性所不允许的㊂然而,如果这一猜想可以得到证实,那么,数学史上认为伽罗瓦定理错了的认识将被修正㊂本文是古证复原或数学实操范式于近现代数学史研究的一例应用[16 19]㊂伽罗瓦对此并没有留下太多的文字,更不用说明确的答案,这就决定了我们必须从间接㊁断裂㊁残缺的原始材料的推理分析入手,复原伽罗瓦对于本原方程概念的实际所指㊂2　本原方程与二重传递方程一个不可约方程是本原的还是非本原的,由它的伽罗瓦群所决定㊂在早期,数学家们以置换群来表述方程的伽罗瓦群,它由作用于方程之根且保持根的有理函数不变的置换全体构成㊂([20],p55)不可约方程的伽罗瓦群是一个传递群([21],p131),一个群作用于一个非空集合传递是指,它存在置换使得该集合中的任意两个元素相互变动㊂进一步地,传递群又可一分为二:一个传递群作用于一个非空集合为非本原群,是指该集合存在相等基数的非平凡互斥子集的一个划分,使得在该群的置换作用下,其中的每个子集仍变为某个子集;相反,若该集合不存在满足如上条件的划分,或者其划分是平凡的,即子集为原集合本身或子集中的元素个数均为1,则这样的传递群就是本原群㊂([22],页112)如果一个不可约方程的伽罗瓦群为本原群,那么,这样的方程就是本原方程;反之,则为非本原方程㊂伽罗瓦虽未证明本原方程与本原群的一一对应,但他在论述中隐含使用了如此假设([11],pp171 191)㊂伽罗瓦将不是非本原的方程称之为本原方程:借助一个m次方程,那些可以分解为m个n次方程的mn次方程称之为非本原方程,这是高斯先生的方程,本原方程是不满足这样一种简化的方程㊂([5],p271)非本原方程所对应的多项式可以通过添加基本域上的一个辅助方程的所有根,在基本域的扩域上分解为次数相等且形式相同的因式乘积①,高斯处理过的分圆方程便是一例㊂([15],pp383 385)相反,一个不可约方程如果不是非本原的,即不满足如上性质,那它就是本原方程,比如素数次不可约方程([23],p295)㊂虽然非本原方程的相反情形可以想象,但对于本原方程,这样的描述仍然是模糊的,①不包含只在分裂域上分解为一次因式的不可约方程,因为,此时其伽罗瓦群作用的集合所满足的划分是平凡的,这样的方程是本原方程㊂07中㊀国㊀科㊀技㊀史㊀杂㊀志45卷因为,我们无法获知本原方程确切的类别和特征,这也为伽罗瓦经由不明确的 本原方程 得到不同的定理而埋下了伏笔㊂传递群的概念可以进一步拓展㊂如果作用于一个非空集合的传递群存在置换,使得该集合中的任意两个不同的元素变动到另外两个不同的元素,那么,这样的传递群就是二重传递群㊂我们已经知道,若一个不可约方程的伽罗瓦群是本原群,则该方程就是本原方程,那么,如果一个不可约方程的伽罗瓦群是二重传递群,它所对应的方程又是怎样的类型呢?假设一个m次多项式f(x)ɪK[x]在域K上不可约,由不可约方程与传递群的对应,知其伽罗瓦群Gal(f,K)传递㊂去掉f(x)=0的任意一根,比如:α1,可以得到扩域K(α1)上的一个多项式f1(x)=f(x)(x-α1)=x m-1+b1x m-2+ +b m-1ɪK(α1)x[].此时,Gal(f,K)二重传递当且仅当f1(x)在K(α1)上不可约㊂([24],p69)伽罗瓦群为二重传递群的不可约方程是这样的类型:在添加其一根所得的基本域的扩域上,它所对应的多项式除去包含此根的一次因式之后仍是不可约的㊂在代数学史上,很多方程的概念都是按照方程与群的对应来命名的,由此,我们不妨将其伽罗瓦群为二重传递群的不可约方程称之为 二重传递方程 (doubly transitive equations)㊂按照现代群论,二重传递群必然是本原群([25],p15),故二重传递方程必然也是本原方程,这是伽罗瓦可能将二重传递方程这一特殊类型的方程理解成一般的本原方程的前提,但问题在于,伽罗瓦的数学中存在二重传递方程吗?他又在什么意义上会将二重传递方程等同于本原方程?在伽罗瓦的卷宗(Dossier.16)中有一片段,伽罗瓦考虑了 方程可以分解为两个或者两个以上因式 的这样一种特殊情形:令U=0是一个方程,且U=VT,V与T是这样的函数,其系数可由原方程系数及添加量有理确定 显然,如果给方程U=0添加V=0的所有根,方程U=0将分解为一些因式,其中一个将是T=0,且其他的将是V的单因式㊂([11],p305)根据拉克鲁瓦(Sylvestre François Lacroix,1765 1843)的‘代数基础“(伽罗瓦读过此书,[11],p5),在伽罗瓦的时代, 单因式 (simple factor)是指形如x-a的一次因式([26],p185)㊂贯通起来,伽罗瓦文本的意思是:给原方程所在的基本域添加其部分根之后,原方程所对应的多项式将在扩域上分解为某个因式与包含这部分根的一些一次因式的乘积,即U=T㊃(x-a)(x-b)而且,伽罗瓦对此情形的讨论建立在这样一个基本假设之上:简单来讲,是在方程无有理因式的情形下㊂事实上,如果我们接受在这种情况下它已经被证明,我们假设一个方程可以分解为两个本身无有理因式的因式㊂([11], p307)伽罗瓦讲 一个方程无有理因式 即指其在基本域上不可约㊂上述文字说明,在伽罗瓦所谓的特殊情形中,原方程在基本域上是不可约的,而且,更重要的是,在添加其部分根之后,它所对应的多项式在基本域的扩域上所分解的各个因式也是不可约的㊂㊀1期杨保强:伽罗瓦定理真的错了吗?17伽罗瓦的特殊情形当然包含最简单的情况:设U=0是基本域K上的一个不可约方程,将它的一个根α添加到基本域K,则在扩域K(α)上,原方程所对应的多项式U分解为两个因式:U=(x-α)㊃T㊂按照伽罗瓦的设定,T在K(α)上也是不可约的㊂此时,伽罗瓦所表述的方程U(x)=0正是域K上的一个二重传递方程㊂伽罗瓦的表述揭示出二重传递方程必然是本原方程㊂因为,按照伽罗瓦的描述,它一定不是非本原的:非本原方程所对应的多项式通过添加一个辅助方程的所有根,会在基本域的某个扩域上分解为次数相等且形式相同的因式乘积,但二重传递方程所对应的多项式通过逐一添加原方程之根,只在分裂域上分解为一次因式的乘积,而在中间任何基本域的扩域上都不存在非本原方程所满足的分解㊂通过以上解读和分析,可以发现,伽罗瓦的手稿不仅包含二重传递方程的等价表述,而且他的表述也暗示了二重传递方程是本原方程㊂在此基础上,伽罗瓦将二重传递方程这种特殊类型的本原方程当成一般意义的本原方程是可能的㊂3　伽罗瓦的 本原方程 :二重传递方程伽罗瓦定理中的 本原方程 是否指代二重传递方程?这是我们评判伽罗瓦定理正确与否的关键,而想要揭开这层历史迷雾,就得使伽罗瓦的真实所想复见于纸上㊂在1832年的 遗书 (The Testamentary Letter)中,伽罗瓦记述了他研究本原方程的概况:最简单的分解就是高斯先生的方法中出现的那些分解㊂无论何时,只要添加一方程的一个根,这个方程将变得可约这些分解是显然的,即使是在方程具体形式中,所以没有必要在这个主题上浪费时间㊂对于一个不能用高斯的方法简化的方程,怎样的分解才是可行的呢?我称那些不能用高斯的方法简化的方程为本原[方程];这些方程并不是真的不能被分解,因为它们甚至可能是根式可解的㊂作为根式可解的本原方程的理论的一个引理,我已经于1830年的6月在费吕萨克通报上发表了关于数论的一个虚数分析㊂同时附之以下定理的证明:1.一个可以根式求解的本原方程,其次数必为p v,p是一个素数㊂2.这类方程的所有置换将具有这样的形式x k,l,m, /x ak+bm+cl+ +f,a1k+b1m+c1l+ +g,其中k,l,m, 是n个指标,每个都取p个值,它们代表所有的根㊂这些指标是模p 而取的,也就是说当给这些指标之一加上p的倍数时,所得的根将是一样的㊂([11],p87)伽罗瓦这次给出了关于素数幂次根式可解的本原方程的正确结果,也就是上述定理2㊂对应于现代群论的表述,该定理是指:一个根式可解的p n(p为素数)次的本原方程的伽罗瓦群同构于n维仿射一般线性群AGL(n,F p)的一个子群㊂27中㊀国㊀科㊀技㊀史㊀杂㊀志45卷值得注意的是,伽罗瓦在原始手稿中划去了这么一句 无论何时,只要添加一方程的一个根,这个方程将变得可约 ,这句话正好成为解开这个历史谜题的关键㊂此句的意思是,每当添加不可约方程f(x)=0(fɪK x[])的一个根x1,该方程将在扩域K(x1)上变得可约㊂这句本身无误,它相当于今天的因式定理的事实:如果域K上的不可约方程f(x)=0有一个根x1,那么,在扩域K(x1)上,x-x1将整除f(x),此即f(x)在K(x1)上可约㊂根据上下文的语境,这一句似乎是在解释非本原方程的概念,因为,在舍略文本的前后,伽罗瓦都说的是可以按照高斯简化方法分解的非本原方程的情形,但对比伽罗瓦关于非本原方程的表述,就会发现,事实并非如此:因为,依照伽罗瓦的表述,非本原方程涉及的是添加另一个辅助方程之所有根的情形,而划去的这句描述的则是添加原方程的一个根的情况㊂伽罗瓦为什么会想到这么一句,而最后又划掉了它?我们知道,二重传递方程的概念有这样两个要点:首先,它在基本域上不可约,但只要通过添加原方程的一个根,它将变得可约,即在基本域的扩域上,它所对应的多项式将分解为包含此根的一次因式与另外一个因式的乘积;其次,第二个因式在新的扩域上也是不可约的㊂显然,伽罗瓦的舍略文本描绘的正是给基本域添加原方程的一个根的情形㊂因伽罗瓦划掉了这句,丢失了文本间的联系,所以伽罗瓦划掉的这句应当意在文外,其中所略需要我们稍作复原㊂事实上,如果顺着伽罗瓦的逻辑,就会发现,他当时想要说明的是:尽管任何不可约方程可以通过添加它的一个根而在基本域的扩域上变得可约,但 本原方程 这种特殊情形则对应的是,除去包含原方程之一根的一次因式的方程,在添加此根所得基本域的扩域上仍然是不可约的(二重传递方程)㊂如果按照这种语言逻辑表述下去,伽罗瓦所要表达的 本原方程 概念就会被解释为二重传递方程,但他在下文给出了关于本原方程可解条件的正确定理,这至少说明,相较之前的 错误 定理,此时,他对于 本原方程 的概念已经有了正确的认识㊂因此,这样的解释如果还写在这里,显然是有悖于下文的真理的㊂所以,伽罗瓦划掉了此句㊂伽罗瓦划去的文本本身是没有错误的,伽罗瓦划掉此句只能是出于与所述事实不符的考虑㊂伽罗瓦划去的这句非常贴近他之前对于二重传递方程的描述,因为仅是二重传递方程才涉及添加原方程之一根的情形㊂故在关于本原方程与非本原方程的语境中,所删文本如果不是指非本原方程的情形,那只能是:伽罗瓦在通过对二重传递方程的刻画来解释他所理解的 本原方程 !因此,伽罗瓦最初所理解的 本原方程 概念对应的是二重传递方程㊂而且,在陈述了本原方程正确的定理之后,伽罗瓦对最初所给的定理作了评注,他表述道: 我在费吕萨克通报中所指明的本原方程根式可解的条件限制性太强㊂ ([11],p 89)伽罗瓦讨论本原方程,却得到一个只在二重传递方程概念下才成立的定理,伽罗瓦自己的评论也暗合了他的定理是将 本原方程 限定于二重传递方程这种特殊的本原方程的理解而得到的㊂所以,伽罗瓦的定理是从二重传递方程的概念出发的㊂综上,伽罗瓦定理中的 本原方程 实际所指应当是二重传递方程㊂而且,伽罗瓦也㊀1期杨保强:伽罗瓦定理真的错了吗?37提到他的定理 例外很少,但还是有一些 ([11],p89)㊂这样,伽罗瓦的定理其实是:伽罗瓦定理.除了9次和25次方程[等],素数幂次[二重传递]方程根式可解的必要条件是,已知它的两个根,其余的根都可以表示为它们的有理函数㊂二重传递方程的伽罗瓦群是二重传递群,该定理正好对应于群论中于佩尔的定理㊂伽罗瓦从二重传递方程出发,得到一个关于二重传递方程的结果,又何错之有呢?或许我们还会有质疑,伽罗瓦明明讨论的是本原方程,却把它理解成二重传递方程去处理,而得到一个二重传递方程的结果,伽罗瓦还是错了,他误解且混淆了本原方程的概念㊂这样的疑问其实带有一种以今度古的倾向㊂首先,这种 错误 已经不再是伽罗瓦定理错误与否的问题;其次,伽罗瓦对基本概念的误解其实也不能算是错误,它只是 名 与 实 的指代和对应问题㊂举个数学史上类似的例子:1853年,克罗内克(Leopold Kronecker,1823 1891)将其伽罗瓦群为循环群的不可约方程称之为 阿贝尔方程 (Abelian equations),但若尔当在1870年建议,应该用 阿贝尔方程 指代阿贝尔所考虑过的更为一般的方程,即其伽罗瓦群是阿贝尓群或交换群的不可约方程㊂克罗内克于1877年接受了若尔当的建议,为作区别,他将之前认识到的循环群所对应的方程改称为 简单阿贝尔方程 ㊂([27],p9)即使这样,博尔萨(Oskar Bolza,1857 1942)在1893年的书评文章中却一仍其旧,把循环群对应的方程称之为 阿贝尔方程 ㊂([28],p103)如此,我们能说后者错了吗?并不能㊂因为,在数学史上,数学家们对于概念的认识本来就是递进的,如同奋身黑夜,每个人的行程长短有限,探见的光亮大小不同,在新的命名方式被认可之前,当然可以用同一个术语指代两种既有融合又互有边界的概念㊂伽罗瓦对本原方程的认识受时代约束本来就是不明确的㊂原因在于,本原方程是通过本原群来界定的,但伽罗瓦对于本原群的理解却是模糊的,比如,他也会把本原群与 拟本原群 等同起来㊂所以,看待伽罗瓦错误与否须得将评判标准转向伽罗瓦究竟从哪种概念出发 本原方程 背后的真实所指才是合理允当的,而这又回到了我们的问题和论证:虽然伽罗瓦定理中的方程名为 本原方程 ,但其实,伽罗瓦却是将二重传递方程当作 本原方程 来理解的,而在二重传递方程的条件下伽罗瓦的定理又是对的,因此,伽罗瓦的定理本身无误㊂4　结论伽罗瓦的定理真的错了吗?数学史界对此历来存有争议㊂将伽罗瓦的定理上升于群论的认识之后,评判伽罗瓦定理正确与否的关键在于伽罗瓦的 本原群 是否是二重传递的㊂不过,由于史料不足,该问题在群的视角下很难得到答案,更何况,伽罗瓦定理的原始表述针对的是 本原方程 ㊂所以,我们通过引入与二重传递群相对应的 二重传递方程 的概念,从方程的角度来考证伽罗瓦定理中的 本原方程 是否指代二重传递方程㊂伽罗瓦对本原方程的表述并不确切,他只提到本原方程不是非本原的㊂在伽罗瓦的卷宗中,我们找到了有关二重传递方程概念的表述,同时,伽罗瓦的描述也暗示了二重传递方程正是本原方程,因为它并不满足非本原方程所满足的分解㊂这两点为伽罗瓦将中㊀国㊀科㊀技㊀史㊀杂㊀志45卷二重传递方程这种特殊类型的本原方程当作一般的本原方程提供了可能㊂伽罗瓦定理中的 本原方程 实际所指乃二重传递方程㊂通过对伽罗瓦的遗书其间舍略文本的语境修复和整体性分析,我们揭示出,伽罗瓦最初的 本原方程 概念是指二重传递方程㊂而且,根据伽罗瓦之后对该定理的评注从旁推断出,伽罗瓦的定理是将他所考虑的 本原方程 限制于二重传递方程这种特殊类型的本原方程得到的㊂从二重传递方程的概念出发,伽罗瓦得到了一个二重传递方程的结果,而且,按照方程与群的对应,如不考虑伽罗瓦未找到的例外(但他承认还有例外),它正好对应群论中的于佩尔定理㊂因此,伽罗瓦的定理并无差错㊂以此,我们修正了数学史上认为伽罗瓦定理错误的观点㊂前人学者所认为的伽罗瓦定理的 错误 实际上是一种被误解的 错误 ,在他们看来,伽罗瓦定理当中的 本原方程 就是我们今天所指的一般意义的本原方程,但根据我们的考证,事实并非如此,伽罗瓦的定理只针对于这种特殊的本原方程 二重传递方程㊂有趣的是,这种被误解的 错误 亦已成为过去数学史的一部分,对后世数学家若尔当的研究以及代数学的发展产生了深远的影响㊂[29 30]图1㊀伽罗瓦定理的认知图数学家的想法,不为载录的,一经过往,便成为历史的一部分,而止于文字的,又未必是我们所理解的真实㊂这就要求我们,在提出或者采信某种数学史观点时应当设身处地地接近古人㊂数学家从不同的概念出发,会得到不同的数学真实,比如,二重传递群的定理㊁本原群的定理,但我们要寻找的历史真实只有一个,那就是:伽罗瓦的定理究竟是二重传递群的定理还是本原群的定理?数学真实不能完全再现历史真实,但它可以引导我们揣思历史真实,而无记述的历史真实可以通过古证复原或数学实操等史学研究方法揭示出来㊂致㊀谢㊀感谢西北大学的曲安京教授对本研究的启发和指导,谢谢京都大学的上野健尔(Kenji Ueno )教授提供了伽罗瓦手稿的相关资料!向匿名审稿人的宝贵意见与建议表达衷心的谢忱!参㊀考㊀文㊀献1㊀Abel N H.Sur la Résolution Algébrique des Equations[A].Sylow L,Lie S(eds.).Oeuvres Complètes de Niels Henrik Abel [C].Christiania:Grøndahl,1881.2㊀Jordan C.Notice sur les Travaux de M.Camille Jordan L appui de sa Candidature L Académie des Sciences[J].47。