This section describes functionality to classify isolated singularities which are analytically hypersurface singularities as defined in the previous subsection Analytically Hypersurface Singularities. The classification is up to analytic isomorphism. The singularities are specified by giving a multivariate polynomial or power series f such that the completion of the local ring of the singularity is isomorphic to the quotient by f of the power series ring in which it lies. The analytic classification comes down to finding nice spaces of relatively simple polynomials (the normal forms) - these form the classification families - such that any power series with a singularity at the origin is analytically equivalent to one of these normal forms. The classification type refers to the particular family that the normal form lies in. Elements of distinct families should not be analytically equivalent. The normal forms within a family should depend on a finite number of continuous parameters and at most finitely many forms within the family may be analytically equivalent to any given form (a finite group of family automorphisms is allowable).
The classification scheme we follow here is that of Arnold as described in Chapter 15 of [AGZV85]. For those singularities that lie in one of the families of this classification, we determine the family and normal form. Note that his classification is a partial classification giving a finite set of multi-indexed, irreducible families occurring in the infinite stratification by Milnor number of the space of all isolated analytic hypersurface singularities up to stable equivalence. However, his classification at least covers all singularities with Milnor number ≤16 and all irreducible families with constant Milnor number that have codimension ≤10 in the full space. It includes much more besides: any of his indexed families contain singularities with an infinite range of Milnor numbers as the (integer) index varies.
Any power series in characteristic != 2 with no constant or linear term can be analytically transformed into a diagonal quadratic part x12 + ... + xr2 plus a power series in the remaining variables with all terms of degree ≥3. Stable equivalence of power series means that the residual degree ≥3 parts are analytically isomorphic.The corank refers to the number of variables not lying in the quadratic part.
Arnold's families are all affine, polynomial families : the normal forms for the family are a fixed polynomial plus a linear combination of a finite number of monomials, all in a subset of the variables and all terms of which are of degree ≥3, which is added to the diagonal quadratic form in the remaining variables. The degree ≥3 part is the essential part and we sometimes use the term normal form to refer to just this component of the full normal form. It is a polynomial in a number of variables equal to the corank of the singularity.
For example, the normal forms of the corank 2 family denoted Jk, 0 by Arnold with k > 2 are the polynomial f = x3 + bx2yk + y3k (b is a constant here) added to an arbitrary linear combination of the k - 2 monomials xy2k + 1, ..., xy3k - 2 and then added to the diagonal quadratic part. So if the number of variables is 3 - let them be x, y, z - we would have p(x, y) + z2, where p(x, y) is f plus the xyr part. If we have 4 variables x, y, z, t, we get p(x, y) + z2 + t2 etc. In our implementation, we actually transform to a normal form where the diagonal quadratic part consists of the initial variables and the essential corank part is in the last variables.
As well as computing the normal form, the user can ask for an explicit analytic isomorphism that transforms the original hypersurface equation to the normal form up to terms of any required precisional degree. If the precision is not specified, the transformation is computed up to terms of a sufficiently high degree to uniquely determine the normal form (i.e. the image of the truncated transformation is the normal form plus higher degree error terms that do not affect the analytic equivalence class). In the many cases of families where there is a natural weighting of the variables associated to the normal forms, the default precision may be different for different variables. This will be described more precisely in the intrinsic descriptions.
Arnold's classification is over an algebraically-closed field. Implementationally, the hypersurfaces are generally given over number fields or finite fields. Although we transform and work over finite extensions when absolutely necessary, we try to avoid the overhead of forming too many field extensions. Thus, we often do not normalise completely when computing normal forms, returning an almost-normal form that can easily be transformed to the full normal form by the user if desired. The major way in which we avoid complete normalisation is in not replacing terms of the form axr with xr by taking an rth root of a and transforming by x |-> a( - 1/r)x. These scaling transformations to reduce a coefficient in the main polynomial to 1 are only applied in cases where no roots need be computed (e.g. we can transform axy3 to xy3 by mapping x to (1/a)x). In particular, the quadratic part of the normal form will be of the form a1x12 + ... + arxr2 for non-zero a1, ..., ar rather than x12 + ... + xr2. Beside avoiding these scalings, there are a few other places where we use a slightly different normal form to Arnold's in order to avoid field extensions. These will be listed in the relevant intrinsic descriptions.
The classification is over an (algebraically-closed) field of characteristic zero. However, for any particular family, the classification process leading to that family and the transformation to the specific normal forms within that family work in all but a small number of characteristics (the precise ones depending on the family). As it is often useful in computational geometry to work modulo a moderate size prime to avoid coefficient blow-up in Gröbner basis operations, we put no restriction on the characteristic of the base fields over which the singularities are defined. We have tried to be very careful in the implementation to work out for precisely which characteristics each (indexed) family does not work and only disallow input at the point where we have reached a family and found that we have a bad characteristic for it. So, if the user is working in a positive characteristic the intrinsics may return a runtime error at some stage. The error message indicates the apparent type of the singularity and warns that it is an invalid family for the characteristic. In valid cases, the polynomial normal forms are the same in the positive characteristic as in characteristic zero and the isomorphic transformation to the result can be computed to an arbitrarily high precision (the finite precision transformations returned are the truncations of genuine analytic isomorphisms in that characteristic).
The input hypersurfaces can be given as exact polynomials or defined by the data records returned by the IsHypersurfaceSingularity intrinsic of the previous subsection. These records allow the computation to any desired precision of the expansion of a power series F that defines the singularity at the origin. This is sufficient to determine the family and normal form. In fact, if the Milnor number of the power series F is a finite number μ (it is finite for any isolated singularity in characteristic zero and for all of our families in the allowable characteristics) then any other power series that is equal to F up to terms of degree μ + 2 or higher, is (formally) analytically isomorphic to F. This gives a general upper degree bound in terms of Milnor number for the degree that expansion up to will determine the singularity up to isomorphism. Actually, the normal forms of the various families usually have highest degree terms of degree much lower than the Milnor number and we are careful in our implementation to only expand as far as necessary to determine the family, normal form and truncated transformation.
All of Arnold's families are of corank ≤4, with the majority of corank 2 and 3. In the initial magma V2.21 release, three of the corank 3 families and two of the more complex corank 2 families remain unimplemented. We hope to add the remaining corank 3 cases for a mid-year release.
The Arnold classification is a fundamental result in singularity theory and the functionality to be able to apply it to magma schemes should be useful in a number of situations. One of our future aims is to use it to speed up the formal desingularisation procedures for the wide range of singular surface models that only contain isolated hypersurface singularities. The normal forms are of a very simple shape - containing only a few monomials in the majority of cases - so the singularities should be much faster to resolve after transformation to normal form.
get_trans: BoolElt Default: false
d: RngIntElt Default: 0
fData: List Default: [**]
milnor_number: RngIntElt Default: -1
The main intrinsic to classify an isolated hypersurface singularity, compute its Arnold normal form and optionally compute to any desired precision an explicit analytic isomorphism to that form.The argument f should be a multivariate polynomial over a field k giving the equation of an affine hypersurface with a singularity at the origin (0, 0, ..., 0). The singularity does not have to be defined by a global hypersurface. It can be an analytically hypersurface singularity as described in the previous subsection Analytically Hypersurface Singularities. To use the intrinsic in this context, the data record fdat returned as the fourth value by IsHypersurfaceSingularity is required. Then f should be the expansion to some finite degree fdeg of the power series F defining the singularity (i.e. F = f + terms of degree ≥fdeg + 1 where f should not be 0 here!) and parameter fData should be set to the two-element magma list [* fdat, fdeg *]. The element fdat will be used to expand f to higher precision if necessary. The intrinsic makes use of the Milnor number of the singularity. If this is already known, the user can pass it in using the parameter milnor_number. This can avoid an expensive recomputation for more complicated singularities. Note, however, that the wrong value passed in will result in runtime errors or invalid return values.
If an explicit analytic isomorphism φ from f to the normal form is also required, the parameter get_trans should be set to true. Let the variables of f be x1, ..., xn and let f0 denote the normal form. Then there is a sequence of n power series in R = k[[x1 ..., xn]] with constant term zero, [s1 ..., sn], such that φ is an isomorphism R -> R, xi |-> si, mapping f to f0. The condition that φ is an isomorphism is equivalent to the matrix of coefficients of the degree 1 terms of the si being invertible. When f0 is in L[x1, ..., xn] for a finite field extension L of k, φ is defined by power series over L. Of course, we cannot represent the power series of the isomorphism exactly, so φ is represented by a magma homomorphism from L[x1, .., xn] (the base change of the parent ring of f to L) to itself which is defined by the sequence of polynomials pi which are the si truncated to a sufficiently high degree.
The precision of the map representing φ is determined as follows. The default precision is the smallest to determine f0 uniquely. The truncated polynomial map defined by the pi maps f to f0 + e, with an error e. The precision of the map is such that e consists of terms of degree higher than those of f0 with respect to a natural weighting of the variables and such that f0 + e is analytically isomorphic to f0 and no other normal form in the same (or different) family. Many of the families in the classification have semiquasihomogeneous normal forms in Arnold's terminology. This means that there is a positive (integer) weighting for the variables for which the fixed polynomial part g of the normal forms (see the introductory section above) is homogeneous (and non-degenerate) of weighted degree d0 and the variable part (if it exists) is a linear combination of monomials of higher degree. In these cases, the default precision is the highest possible weighted degree of the monomials in the variable part (or d0 if there is no variable part). What this means is that there might be much lower ordinary degree terms in the transformation error e for higher weight variables. This convention may seem a little awkward at first but it seems to be the most natural thing in terms of the way that these cases are processed by the code. For non-semiquasihomogeneous cases or when parameter d is used (see below), the precision is always in terms of the ordinary degree of monomials.
If parameter d is set to a positive value (its default is 0), φ will be computed to a precision (in terms of ordinary degree) which is the maximum of d and the default (ordinary) degree needed to determine the normal form uniquely. That is, all terms in the error e are guaranteed to be of ordinary degree greater than d.
The first return value is true if the singularity lies in one of the families of Arnold's classification and false otherwise. If true, the second return value is the normal form f0 of f and the third is a string giving the name of the family. If parameter get_trans is true, then a fourth return value is the polynomial map giving the (truncated) isomorphism from f to f0 as described above.
As discussed in the introduction, the base field k can be of arbitrary characteristic. The only restriction on its magma type is that it must be a field for which finite separable extensions are constructible. A runtime error will occur if it turns out that the shape of the singularity is of a family type for which the characteristic of k is invalid. The apparent family type will be given in the error message. Characteristic 2 will lead to an error in virtually (but not quite) all cases. In particular, an error will result for characteristic 2 fields whenever f has a non-trivial quadratic part. If the singularity is not isolated, this will be discovered when the Milnor number of f is computed (it will be infinite) and false will be returned.
We allow a non-singular point, which we class as a `singularity' of type A0 for which the normal form is x1.
All of the families in the version of Arnold's classification that we use have corank ≤4 (see intro). The only corank 0 singularity is the A1 singularity whose normal form is x12 + ... + xn2. As discussed in the introduction, to avoid excessive field extensions, we do not scale the variables in our transformations to normal form in order to make coefficients 1, so the normal form returned will actually be a1x12 + ... + anxn2 for some non-zero constants ai.
The corank 1 singularities are the Ar, r ≥2. The description strings for these are "A2", "A3" etc. The normal form for these are a1x12 + ... + an - 1xn - 12 + anxnr + 1 where, again, we do not scale the variables to make all of the ai equal to 1.
Corank 2 singularities are described below for the Corank2Case intrinsic. That intrinsic is used by this one but it has also been made available for the user to call directly. There are 3 families of corank 2 singularities that have not yet been covered.
Code for corank 3 in the initial release of V2.21 does not cover all of the families, although we hope to complete it in a mid-year release. The families not yet included lead to a "not yet available" runtime error.
The only corank 4 case in the classification (sometime not included) is O16 (descriptor: "O16"). The normal form for these is the diagonal quadratic part added to a non-degenerate homogeneous cubic in 4 variables (non-degenerate meaning that it defines a non-singular projective cubic surface) and a 1-dimensional `variable' part (maybe zero) given by a homogeneous degree 4 polynomial in the 4 variables. The standard normalisation here is to transform the homogeneous cubic to the form x3 + y3 + z3 + t3 + (ax + by + cz + dt)3 for constants a, b, c, d and the variable part is then given by a scaler multiple of the monomial xyzt. Again, to avoid a possibly large field extension and because not all non-singular cubics can be linearly transformed into this form, we currently take a lazy approach and do not attempt to normalise the cubic part into any particular shape. If this case occurs and the user really wants to try to get a transformation to the standard form, he may use intrinsic PentahedronIdeal to construct a linear transformation in the last 4 variables to move the cubic part to the correct shape. Applying this to the full normal form returned will give the Arnold normal form.
The implementation follows the decision procedure laid out in Chapter 16 of [AGZV85]. The transformations to full normal form given the lowest degree part are effected in a uniform manner via 3 functions that respectively deal with the semiquasihomogeneous cases, the non-sqh cases that satisfy condition A for a Newton filtration (see page 212 of [AGZV85]), and finally the non sqh-cases for which the spectral sequence analysis of Chapter 14 of [AGZV85] applies. In many of the non-sqh cases, the sqh function is first applied to get to the fixed part of the normal form and then the relevant other function is used to find the variable part and kill off other terms.
get_trans: BoolElt Default: false
milnor_number: RngIntElt Default: -1
d: RngIntElt Default: 0
fData: List Default: [**]
This is a direct version of the previous general intrinsic that only deals with the non-quadratic part of hypersurface singularities of corank 2 as defined in the introduction. Argument f should be a polynomial in two variables with all terms of degree at least 3 which is a defining equation for the singularity at the origin (0, 0). The parameter milnor_number if given a positive value can be used to input the known Milnor number of f to save a small amount of computation time. The remaining parameters and return values have the same meaning as for the general intrinsic. In particular fData can be used when f is the truncation of a power series F that gives the exact defining equation for the (analytically) hypersurface singularity.The families of singularities covered by this case are, in Arnold's notation: Dk, k ≥4; E6k, E6k + 1, E6k + 2, k ≥1; Jk, i, k ≥2, i ≥0; Xk, i, k, i ≥1; Y1r, s, r ≥s ≥1; Z6k + 5, Z6k + 6, Z6k + 7, k ≥2; Zk, i, k ≥1, i ≥0; W12k, W12k + 1, W12k + 5, W12k + 6, k ≥1; Wk, i, k ≥1, i ≥0; W^#k, i, k ≥1, i ≥1; N16.
The description strings are the obvious translations of these: e.g., for Y12, 1, we return "Y^1_{2,1}". There are 3 multi-indexed families with atypical product normal forms that we have not yet finished implementing: Ykr, s, Zkr, Zki, j for k ≥2. Currently, if the analysis leads to one of these families, a runtime error results informing the user of the precise family type but warning that the exact normal form (and transformation) cannot currently be computed.
The expressions for normal forms over an algebraically-closed field for each family can be found in Chapter 15 of [AGZV85]. As stated in the introduction, in order to avoid unnecessary field extensions, the normal forms we return are either not fully normalised or are slightly different to those given there. Besides the ubiquitous avoidance of scaling a variable x to transform a term of the form axr to xr with r ≥2, the following variants are used.
For Jk, 0, k ≥2, instead of taking the quasihomogeneous part of the normal form to be ax3 + bx2yk + cy3k, we instead use ax3 + bxy2k + cy3k (a, b, c != 0) except in the one case when transformation to this by x |-> x + uyk leads to c = 0. Then we use the usual form ax3 + bx2yk + cy3k with 2b3=27a2c (this gives one isomorphism class over the algebraic closure on scaling x and y appropriately). The monomial basis for the variable part of the normal form remains xy2k + 1, ..., xy3k - 2.
Similarly, for Zk, 0, k ≥1, instead of using y(ax3 + bx2yk + 1k + cy3k + 3) for the quasihomogeneous part, we use y(ax3 + bxy2k + 2k + cy3k + 3) except in the same unique case as above. The monomial basis for the variable part of the normal form remains xy2k + 4, ..., xy3k + 3.
Singularities of type N16 have a normal form which is a homogeneous polynomial of degree 5 in x, y that is non-degenerate (i.e. it splits into distinct linear factors over the algebraic closure of the base field) plus a higher degree part which lies in a one-dimensional subspace of homogeneous polynomials of degree 6. The normal form given by Arnold over an algebraically-closed field involves a linear transformation of the homogeneous part to a polynomial of the form x4y + ax3y2 + bx2y3 + xy4, in which case the full normal form is this degree 5 polynomial plus a term of the form cx3y3. As the transformation of the degree 5 part generally involves a field extension of degree 20 (2 roots of a degree 5 polynomial need to be added to the base field), we do not make this transformation. We leave the homogeneous degree 5 part f5 as it first occurs and the full normal form returned will be of the form f5 + f6 where f6 is a homogeneous degree 6 part that lies in a one-dimensional subspace, depending on f5, of degree 6 x, y-polynomials. As for the O16 case in the general intrinsic, the user may determine the Arnold normal form by finding the linear translation on f5 to map it to standard form and apply this transformation to f5 + f6.
get_trans: BoolElt Default: false
milnor_number: RngIntElt Default: -1
d: RngIntElt Default: 0
fData: List Default: [**]
This is the direct version of the general NormalFormOfHypersurfaceSingularity intrinsic that only deals with the non-quadratic part of hypersurface singularities of corank 3 as defined in the introduction. Argument f should be a polynomial in three variables with all terms of degree at least 3 which is a defining equation for the singularity at the origin (0, 0, 0). The parameters are exactly the same as for the above Corank2 intrinsic.The families of singularities covered by this case are, in Arnold's notation: Pk=T3, 3, k - 5, k ≥8; Rp, q=T3, p, q, 4 ≤p ≤q; Tp, q, r, 4 ≤p ≤q ≤r; Q6k + 4, Q6k + 5, Q6k + 6, k ≥1; Qk, i, k ≥2, i ≥0; S12k - 1, S12k, S12k + 4, S12k + 5, k ≥1; Sk, i, k ≥1, i ≥0; S^#k, i, k, i ≥1; U12k, U12k + 4, k ≥1; Uk, i, k ≥1, i ≥0; V1, i, V^#1, i, i ≥0.
Again, the description strings contained in the third return value are obvious text translations of these. In the first release covering corank 3 singularities (V2.21, initial release), some of the later families have not yet been covered and these will give a "Not yet available" runtime error. These cases should be implemented and exported in a mid-year release. We do not cover the S * , U * and V * families that Arnold gives as particular classes with specific invariants but without precise normal forms. A singularity falling into one of these classes will result in a first return value of false.
Apart from the usual avoidance of root-scaling variables to transform, e.g., axr to xr when r ≥2, the only major difference between the normal forms computed and Arnold's versions is for P8. A P8 singularity is one whose equation can be transformed into a homogeneous cubic (in 3 variables) that is non-singular as the equation for a projective (genus 1) curve. Arnold uses the normal form x3 + y3 + z3 + axyz where a is the family parameter. We transform to a non-singular cubic but do not convert it to this specific form, partly to avoid the necessary field extensions. the user may use magma functions for elliptic curves to find the explicit transformation to Arnold's form. Two P8 singularities given by homogeneous cubics are isomorphic if and only if the elliptic j-invariants are the same, where each cubic is considered as an elliptic curve by choosing any projective point on it as the origin. If the chosen origin is actually one of the 9 inflexion points of the cubic then an elliptic curve isomorphism between the two will be a linear transformation in the base variables which therefore gives an analytic isomorphism between the two singularities.
> Q := Rationals(); > P<x,y,z> := PolynomialRing(Q,3,"grevlex"); > f := z^8-3*z^7-3*x*z^5+3*y*z^5+z^6+3*x*z^4-3*y*z^4+2*y^4+3*x^2*z^2-6*x*y*z^2+ > 3*y^2*z^2+x^3-3*x^2*y+7*x*y^2-y^3-4*y^2*z+2*x^2-4*x*z+2*z^2; > boo,f0,typ := NormalFormOfHypersurfaceSingularity(f); > boo; f0; typ; true z^8 + 3*y*z^5 - y^3 + 2*x^2 E13The normal form has highest term of degree 8. Now, call the intrinsic again to get the transformation that takes f to f0 up to terms of degree at least 9.
> _,_,_,mp := NormalFormOfHypersurfaceSingularity(f : get_trans := true, d := 8); > mp_err := mp(f)-f0; > Min([TotalDegree(t) : t in Terms(mp_err)]); 9 5440For the second example, we take the singularity defined by a two-variable polynomial containing no terms of degree ≤3. Here, we can use the Corank2Case intrinsic directly. The singularity is of type Z1, 1. Although the polynomial may define an irreducible curve (as in this example), the singularity is non-normal and analytically reducible. The defining polynomial splits into a product of 2 irreducible factors in the power series ring. This can be seen from the normal form which is divisible by the linear factor y.
> P<x,y> := PolynomialRing(Rationals(),2,"grevlex");
> f := y^12-6*x*y^10+15*x^2*y^8+x^9+9*x^8*y+36*x^7*y^2+84*x^6*y^3+
> 126*x^5*y^4+126*x^4*y^5+64*x^3*y^6+36*x^2*y^7+9*x*y^8+y^9+2*x^8+
> 16*x^7*y+56*x^6*y^2+112*x^5*y^3+155*x^4*y^4+112*x^3*y^5+56*x^2*y^6+
> 16*x*y^7+2*y^8-6*x^5*y^2-x^3*y^4-3*x^2*y^5-4*x*y^6-2*y^7+x^6+2*x^4*y^2+
> 6*x^3*y^3+9*x^2*y^4+5*x*y^5-x^5-3*x^4*y-6*x^3*y^2-4*x^2*y^3+x^4+x^3*y;
> IsIrreducible(f);
true
> boo,f0,typ := Corank2Case(f);
> boo; f0; typ;
true
y^9 + 2*y^8 - x^2*y^3 + x^3*y
Z_{1,1}
As a final example, we consider an E7 singularity p on a degree 6
complete intersection surface in (P)4. Here, we cannot find a
polynomial giving a global hypersurface defining p by simple elimination
of a variable. However, p is analytically a hypersurface singularity
(as all E7 singularities are) and we use the
IsHypersurfaceSingularity intrinsic to give f, the truncation to degree
3 of a power series F defining p analytically, and the fdat data object
that allows expansion of F to higher degree.
Calling the main intrinsic with just f as an argument leads to a negative result (in fact, it is clear by inspection that f has a non-isolated singularity at the origin). A second call with the fData parameter included, however, leads to the correct result.
> P4<x,y,z,t,u> := ProjectiveSpace(Q,4); > X := Scheme(P4,[x^2+y*t+t^2,z^3+x*t*u+y^2*u-t*u^2]); > p := X![0,0,0,0,1]; > _,f,_,fdat := IsHypersurfaceSingularity(p,3); > R<a,b,c> := Parent(f); > f; b^3 - c^3 + c^2 > NormalFormOfHypersurfaceSingularity(f); false > boo,f0,typ := NormalFormOfHypersurfaceSingularity(f : fData := [*fdat,3*]); > boo; f0; typ; true b*c^3 + b^3 + a^2 E7