In FinInG we provide some basic functionality for algebraic varieties defined over finite fields. The algebraic varieties in FinInG are defined by a list of multivariate polynomials over a finite field, and an ambient geometry. This ambient geometry is either a projective space, and then the algebraic variety is called a projective variety, or an affine geometry, and then the algebraic variety is called an affine variety. In this chapter we give a brief overview of the features of FinInG concerning these two types of algebraic varieties. The package FinInG also contains the Veronese varieties VeroneseVariety (11.7-1), the Segre varieties SegreVariety (11.6-1) and the Grassmann varieties GrassmannVariety (11.8-1); three classical projective varieties. These varieties have an associated geometry map (the VeroneseMap (11.7-3), SegreMap (11.6-3) and GrassmannMap (11.8-3)) and FinInG also provides some general functionality for these.
An algebraic variety in FinInG is an algebraic variety in a projective space or affine space, defined by a list of polynomials over a finite field.
‣ AlgebraicVariety( space, pring, pollist ) | ( operation ) |
‣ AlgebraicVariety( space, pollist ) | ( operation ) |
Returns: an algebraic variety
The argument space is an affine or projective space over a finite field F, the argument pring is a multivariate polynomial ring defined over (a subfield of) F, and pollist is a list of polynomials in pring. If the space is a projective space, then pollist needs to be a list of homogeneous polynomials. In FinInG there are two types of projective varieties: projective varieties and affine varieties. The following operations apply to both types.
gap> F:=GF(9); GF(3^2) gap> r:=PolynomialRing(F,4); GF(3^2)[x_1,x_2,x_3,x_4] gap> pg:=PG(3,9); ProjectiveSpace(3, 9) gap> f1:=r.1*r.3-r.2^2; x_1*x_3-x_2^2 gap> f2:=r.4*r.1^2-r.4^3; x_1^2*x_4-x_4^3 gap> var:=AlgebraicVariety(pg,[f1,f2]); Projective Variety in ProjectiveSpace(3, 9) gap> DefiningListOfPolynomials(var); [ x_1*x_3-x_2^2, x_1^2*x_4-x_4^3 ] gap> AmbientSpace(var); ProjectiveSpace(3, 9)
‣ DefiningListOfPolynomials( var ) | ( attribute ) |
Returns: a list of polynomials
The argument var is an algebraic variety. This attribute returns the list of polynomials that was used to define the variety var.
‣ AmbientSpace( var ) | ( attribute ) |
Returns: an affine or projective space
The argument var is an algebraic variety. This attribute returns the affine or projective space in which the variety var was defined.
‣ PointsOfAlgebraicVariety( var ) | ( operation ) |
‣ Points( var ) | ( operation ) |
Returns: a list of points
The argument var is an algebraic variety. This operation returns the list of points of the AmbientSpace (11.1-3) of the algebraic variety var whose coordinates satisfy the DefiningListOfPolynomials (11.1-2) of the algebraic variety var.
gap> F:=GF(9); GF(3^2) gap> r:=PolynomialRing(F,4); GF(3^2)[x_1,x_2,x_3,x_4] gap> pg:=PG(3,9); ProjectiveSpace(3, 9) gap> f1:=r.1*r.3-r.2^2; x_1*x_3-x_2^2 gap> f2:=r.4*r.1^2-r.4^3; x_1^2*x_4-x_4^3 gap> var:=AlgebraicVariety(pg,[f1,f2]); Projective Variety in ProjectiveSpace(3, 9) gap> points:=Points(var); <points of Projective Variety in ProjectiveSpace(3, 9)> gap> Size(points); 28 gap> iter := Iterator(points); <iterator> gap> for i in [1..4] do > x := NextIterator(iter); > Display(x); > od; [1...] [1..1] [1..2] [111.]
‣ Iterator( pts ) | ( operation ) |
Returns: an iterator
The argument pts is the set of PointsOfAlgebraicVariety (11.1-4) of an algebraic variety var. This operation returns an iterator for the points of an algebraic variety.
11.1-6 \in‣ \in( x, var ) | ( operation ) |
‣ \in( x, pts ) | ( operation ) |
Returns: true or false
The argument x is a point of the AmbientSpace (11.1-3) of an algebraic variety AlgebraicVariety (11.1-1). This operation also works for a point x and the collection pts returned by PointsOfAlgebraicVariety (11.1-4).
A projective variety in FinInG is an algebraic variety in a projective space defined by a list of homogeneous polynomials over a finite field.
‣ ProjectiveVariety( pg, pring, pollist ) | ( operation ) |
‣ ProjectiveVariety( pg, pollist ) | ( operation ) |
‣ AlgebraicVariety( pg, pring, pollist ) | ( operation ) |
‣ AlgebraicVariety( pg, pollist ) | ( operation ) |
Returns: a projective algebraic variety
gap> F:=GF(9); GF(3^2) gap> r:=PolynomialRing(F,4); GF(3^2)[x_1,x_2,x_3,x_4] gap> pg:=PG(3,9); ProjectiveSpace(3, 9) gap> f1:=r.1*r.3-r.2^2; x_1*x_3-x_2^2 gap> f2:=r.4*r.1^2-r.4^3; x_1^2*x_4-x_4^3 gap> var:=AlgebraicVariety(pg,[f1,f2]); Projective Variety in ProjectiveSpace(3, 9) gap> DefiningListOfPolynomials(var); [ x_1*x_3-x_2^2, x_1^2*x_4-x_4^3 ] gap> AmbientSpace(var); ProjectiveSpace(3, 9)
Quadrics (QuadraticVariety (11.3-2)) and Hermitian varieties (HermitianVariety (11.3-1)) are projective varieties that have the associated quadratic or hermitian form as an extra attribute installed. Furthermore, we provide a method for PolarSpace taking as an argument a projective algebraic variety.
‣ HermitianVariety( pg, pring, pol ) | ( operation ) |
‣ HermitianVariety( pg, pol ) | ( operation ) |
‣ HermitianVariety( n, F ) | ( operation ) |
‣ HermitianVariety( n, q ) | ( operation ) |
Returns: a hermitian variety in a projective space
For the first two methods, the argument pg is a projective space, pring is a polynomial ring, and pol is polynomial. For the third and fourth variations, the argument n is an integer, the argument F is a finite field, and the argument q is a prime power. These variations of the operation return the hermitian variety associated to the standard hermitian form in the projective space of dimension n over the field F of order q.
gap> F:=GF(25); GF(5^2) gap> r:=PolynomialRing(F,3); GF(5^2)[x_1,x_2,x_3] gap> x:=IndeterminatesOfPolynomialRing(r); [ x_1, x_2, x_3 ] gap> pg:=PG(2,F); ProjectiveSpace(2, 25) gap> f:=x[1]^6+x[2]^6+x[3]^6; x_1^6+x_2^6+x_3^6 gap> hv:=HermitianVariety(pg,f); Hermitian Variety in ProjectiveSpace(2, 25) gap> AsSet(List(Lines(pg),l->Size(Filtered(Points(l),x->x in hv)))); [ 1, 6 ] gap> hv:=HermitianVariety(5,4); Hermitian Variety in ProjectiveSpace(5, 4) gap> hps:=PolarSpace(hv); <polar space in ProjectiveSpace( 5,GF(2^2)): x_1^3+x_2^3+x_3^3+x_4^3+x_5^3+x_6^3=0 > gap> hf:=SesquilinearForm(hv); < hermitian form > gap> PolynomialOfForm(hf); x_1^3+x_2^3+x_3^3+x_4^3+x_5^3+x_6^3
‣ QuadraticVariety( pg, pring, pol ) | ( operation ) |
‣ QuadraticVariety( pg, pol ) | ( operation ) |
‣ QuadraticVariety( n, F, type ) | ( operation ) |
‣ QuadraticVariety( n, q, type ) | ( operation ) |
‣ QuadraticVariety( n, F ) | ( operation ) |
‣ QuadraticVariety( n, q ) | ( operation ) |
Returns: a quadratic variety in a projective space
In the first two methods, the argument pg is a projective space, pring is a polynomial ring, and pol is a polynomial. The latter four return a standard non-degenerate quadric. The argument n is a projective dimension, F is a field, and q is a prime power that gives just the order of the defining field. If the type is given, then it will return a quadric of a particular type as follows:
| variety | standard form | characteristic p | proj. dim. | type |
| hyperbolic quadric | X0 X1 + ... + Xn-1Xn | p ≡ 3 mod 4 or p=2 | odd | "hyperbolic", "+", or "1" |
| hyperbolic quadric | 2(X0 X1 + ... + Xn-1Xn ) | p ≡ 1 mod 4 | odd | "hyperbolic", "+", or "1" |
| parabolic quadric | X02 + X1 X2 + ... + Xn-1Xn | p ≡ 1,3 mod 8 or p=2 | even | "parabolic", "o", or "0" |
| parabolic quadric | t(X02 + X1 X2 + ... + Xn-1Xn), t a primitive element of GF(p) | p ≡ 5,7 mod 8 | even | "parabolic", "o", or "0" |
| elliptic quadric | X02 + X12 + X2 X3 + ... + Xn-1Xn | p ≡ 3 mod 4 | odd | "elliptic", "-", or "-1" |
| elliptic quadric | X02 + tX12 + X2 X3 + ... + Xn-1Xn, t a primitive element of GF(p) | p ≡ 1 mod 4 | odd | "elliptic", "-", or "-1" |
| elliptic quadric | X02 + X0X1 + d X12 + X2 X3 + ... + Xn-1Xn , Tr(d) = 1 | even | odd | "elliptic", "-", or "-1" |
If no type is given, and only the dimension and field/field order are given, then it is assumed that the dimension is even and the user wants a standard parabolic quadric.
gap> F:=GF(5); GF(5) gap> r:=PolynomialRing(F,4); GF(5)[x_1,x_2,x_3,x_4] gap> x:=IndeterminatesOfPolynomialRing(r); [ x_1, x_2, x_3, x_4 ] gap> pg:=PG(3,F); ProjectiveSpace(3, 5) gap> Q:=x[2]*x[3]+x[4]^2; x_2*x_3+x_4^2 gap> qv:=QuadraticVariety(pg,Q); Quadratic Variety in ProjectiveSpace(3, 5) gap> AsSet(List(Planes(pg),z->Size(Filtered(Points(z),x->x in qv)))); [ 1, 6, 11 ] gap> qf:=QuadraticForm(qv); < quadratic form > gap> Display(qf); Quadratic form Gram Matrix: . . . . . . 1 . . . . . . . . 1 Polynomial: [ [ x_2*x_3+x_4^2 ] ] gap> IsDegenerateForm(qf); #I Testing degeneracy of the *associated bilinear form* true gap> qv:=QuadraticVariety(3,F,"-"); Quadratic Variety in ProjectiveSpace(3, 5) gap> PolarSpace(qv); <polar space in ProjectiveSpace(3,GF(5)): x_1^2+Z(5)*x_2^2+x_3*x_4=0 > gap> Display(last); <polar space of rank 3 over GF(5)> Non-singular elliptic quadratic form Gram Matrix: 1 . . . . 2 . . . . . 1 . . . . Polynomial: [ [ x_1^2+Z(5)*x_2^2+x_3*x_4 ] ] Witt Index: 1 Bilinear form Gram Matrix: 2 . . . . 4 . . . . . 1 . . 1 . gap> qv:=QuadraticVariety(3,F,"+"); Quadratic Variety in ProjectiveSpace(3, 5) gap> Display(last); Quadratic Variety in ProjectiveSpace(3, 5) Polynomial: [ Z(5)*x_1*x_2+Z(5)*x_3*x_4 ]
‣ QuadraticForm( var ) | ( attribute ) |
Returns: a quadratic form
When the argument var is a QuadraticVariety (11.3-2), this returns the associated quadratic form.
‣ SesquilinearForm( var ) | ( attribute ) |
Returns: a hermitian form
If the argument var is a HermitianVariety (11.3-1), this returns the associated hermitian form.
‣ PolarSpace( var ) | ( operation ) |
the argument var is a projective algebraic variety. When its list of defining polynomial contains exactly one polynomial, depending on its degree, the operation QuadraticFormByPolynomial or HermitianFormByPolynomial is used to compute a quadratic form or a hermitian form. These operations check whether this is possible, and produce an error message if not. If the conversion is possible, then the appropriate polar space is returned.
gap> f := GF(25); GF(5^2) gap> r := PolynomialRing(f,4); GF(5^2)[x_1,x_2,x_3,x_4] gap> ind := IndeterminatesOfPolynomialRing(r); [ x_1, x_2, x_3, x_4 ] gap> eq1 := Sum(List(ind,t->t^2)); x_1^2+x_2^2+x_3^2+x_4^2 gap> var := ProjectiveVariety(PG(3,f),[eq1]); Projective Variety in ProjectiveSpace(3, 25) gap> PolarSpace(var); <polar space in ProjectiveSpace(3,GF(5^2)): x_1^2+x_2^2+x_3^2+x_4^2=0 > gap> eq2 := Sum(List(ind,t->t^4)); x_1^4+x_2^4+x_3^4+x_4^4 gap> var := ProjectiveVariety(PG(3,f),[eq2]); Projective Variety in ProjectiveSpace(3, 25) gap> PolarSpace(var); Error, <poly> does not generate a Hermitian matrix called from GramMatrixByPolynomialForHermitianForm( pol, gf, n, vars ) called from HermitianFormByPolynomial( pol, pring, n ) called from HermitianFormByPolynomial( eq, r ) called from <function "unknown">( <arguments> ) called from read-eval loop at line 16 of *stdin* you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk> quit; gap> eq3 := Sum(List(ind,t->t^6)); x_1^6+x_2^6+x_3^6+x_4^6 gap> var := ProjectiveVariety(PG(3,f),[eq3]); Projective Variety in ProjectiveSpace(3, 25) gap> PolarSpace(var); <polar space in ProjectiveSpace(3,GF(5^2)): x_1^6+x_2^6+x_3^6+x_4^6=0 >
An affine variety in FinInG is an algebraic variety in an affine space defined by a list of polynomials over a finite field.
‣ AffineVariety( ag, pring, pollist ) | ( operation ) |
‣ AffineVariety( ag, pollist ) | ( operation ) |
‣ AlgebraicVariety( ag, pring, pollist ) | ( operation ) |
‣ AlgebraicVariety( ag, pollist ) | ( operation ) |
Returns: an affine algebraic variety
The argument ag is an affine space over a finite field F, the argument pring is a multivariate polynomial ring defined over (a subfield of) F, and pollist is a list of polynomials in pring.
A geometry map is a map from a set of elements of a geometry to a set of elements of another geometry, which is not necessarily a geometry morphism. Examples are the SegreMap (11.6-3), the VeroneseMap (11.7-3), and the GrassmannMap (11.8-3).
‣ Source( gm ) | ( operation ) |
Returns: the source of a geometry map
The argument gm is a geometry map.
‣ Range( gm ) | ( operation ) |
Returns: the range of a geometry map
The argument gm is a geometry map.
‣ ImageElm( gm, x ) | ( operation ) |
Returns: the image of an element under a geometry map
The argument gm is a geometry map, the element x is an element of the Source (11.5-1) of the geometry map gm.
‣ ImagesSet( gm, elms ) | ( operation ) |
Returns: the image of a subset of the source under a geometry map
The argument gm is a geometry map, the elements elms is a subset of the Source (11.5-1) of the geometry map gm.
11.5-5 \^‣ \^( x, gm ) | ( operation ) |
Returns: the image of an element of the source under a geometry map
The argument gm is a geometry map, the element x is an element of the Source (11.5-1) of the geometry map gm.
A Segre variety in FinInG is a projective algebraic variety in a projective space over a finite field. The set of points that lie on this variety is the image of the Segre map.
‣ SegreVariety( listofpgs ) | ( operation ) |
‣ SegreVariety( listofdims, field ) | ( operation ) |
‣ SegreVariety( pg1, pg2 ) | ( operation ) |
‣ SegreVariety( d1, d2, field ) | ( operation ) |
‣ SegreVariety( d1, d2, q ) | ( operation ) |
Returns: a Segre variety
The argument listofpgs is a list of projective spaces defined over the same finite field, say [PG(n1 -1,q), PG(n2 -1,q), ..., PG(nk -1,q)]. The operation also takes as input the list of dimensions (listofdims) and a finite field field (e.g. [n1, n2, ..., nk,GF(q)]). A Segre variety with only two factors (k=2), can also be constructed using the operation with two projective spaces pg1 and pg2 as arguments, or with two dimensions d1, d2, and a finite field field(or a prime power q). The operation returns a projective algebraic variety in the projective space of dimension n1 n2 ... nk-1.
‣ PointsOfSegreVariety( sv ) | ( operation ) |
‣ Points( sv ) | ( operation ) |
Returns: the points of a Segre variety
The argument sv is a Segre variety. This operation returns a set of points of the AmbientSpace (11.1-3) of the Segre variety. This set of points corresponds to the image of the SegreMap (11.6-3).
‣ SegreMap( listofpgs ) | ( operation ) |
‣ SegreMap( listofdims, field ) | ( operation ) |
‣ SegreMap( pg1, pg2 ) | ( operation ) |
‣ SegreMap( d1, d2, field ) | ( operation ) |
‣ SegreMap( d1, d2, q ) | ( operation ) |
‣ SegreMap( sv ) | ( operation ) |
Returns: a geometry map
The argument listofpgs is a list of projective spaces defined over the same finite field, say [PG(n1 -1,q), PG(n2 -1,q), ..., PG(nk -1,q)]. The operation also takes as input the list of dimensions (listofdims) and a finite field field (e.g. [n1, n2, ..., nk,GF(q)]). A Segre map with only two factors (k=2), can also be constructed using the operation with two projective spaces pg1 and pg2 as arguments, or with two dimensions d1, d2, and a finite field field(or a prime power q). The operation returns a function with domain the product of the point sets of projective spaces in the list [PG(n1 -1,q), PG(n2 -1,q), ..., PG(nk -1,q)] and image the set of points of the Segre variety (PointsOfSegreVariety (11.6-2)) in the projective space of dimension n1 n2 ... nk-1. When a Segre variety sv is given as input, the operation returns the associated Segre map.
gap> sv:=SegreVariety(2,2,9); Segre Variety in ProjectiveSpace(8, 9) gap> sm:=SegreMap(sv); Segre Map of [ <points of ProjectiveSpace(2, 9)>, <points of ProjectiveSpace(2, 9)> ] gap> cart1:=Cartesian(Points(PG(2,9)),Points(PG(2,9)));; gap> im1:=ImagesSet(sm,cart1);; gap> Span(im1); ProjectiveSpace(8, 9) gap> l:=Random(Lines(PG(2,9))); <a line in ProjectiveSpace(2, 9)> gap> cart2:=Cartesian(Points(l),Points(PG(2,9)));; gap> im2:=ImagesSet(sm,cart2);; gap> Span(im2); <a proj. 5-space in ProjectiveSpace(8, 9)> gap> x:=Random(Points(PG(2,9))); <a point in ProjectiveSpace(2, 9)> gap> cart3:=Cartesian(Points(PG(2,9)),Points(x));; gap> im3:=ImagesSet(sm,cart3);; gap> pi:=Span(im3); <a plane in ProjectiveSpace(8, 9)> gap> AsSet(List(Points(pi),y->y in sv)); [ true ]
‣ Source( sm ) | ( operation ) |
Returns: the source of a Segre map
The argument sm is a SegreMap (11.6-3). This operation returns the cartesian product of the list consisting of the pointsets of the projective spaces that were used to construct the SegreMap (11.6-3).
A Veronese variety in FinInG is a projective algebraic variety in a projective space over a finite field. The set of points that lie on this variety is the image of the Veronese map.
‣ VeroneseVariety( pg ) | ( operation ) |
‣ VeroneseVariety( n-1, field ) | ( operation ) |
‣ VeroneseVariety( n-1, q ) | ( operation ) |
Returns: a Veronese variety
The argument pg is a projective space defined over a finite field, say PG(n-1,q). The operation also takes as input the dimension and a finite field field (e.g. [n-1,q]). The operation returns a projective algebraic variety in the projective space of dimension (n^2+n)/2-1, known as the (quadratic) Veronese variety. It is the image of the map (x0,x1,...,xn)→ (x02,x0x1,...,x0xn,x12,x1x2,...,x1xn,...,xn2)
‣ PointsOfVeroneseVariety( vv ) | ( operation ) |
‣ Points( vv ) | ( operation ) |
Returns: the points of a Veronese variety
The argument vv is a Veronese variety. This operation returns a set of points of the AmbientSpace (11.1-3) of the Veronese variety. This set of points corresponds to the image of the VeroneseMap (11.7-3).
‣ VeroneseMap( pg ) | ( operation ) |
‣ VeroneseMap( n-1, field ) | ( operation ) |
‣ VeroneseMap( n-1, q ) | ( operation ) |
‣ VeroneseMap( vv ) | ( operation ) |
Returns: a geometry map
The argument pg is a projective space defined over a finite field, say PG(n-1,q). The operation also takes as input the dimension and a finite field field (e.g. [n-1,q]). The operation returns a function with domain the product of the point set of the projective space PG(n-1,q) and image the set of points of the Veronese variety (PointsOfVeroneseVariety (11.7-2)) in the projective space of dimension (n^2+n)/2-1. When a Veronese variety vv is given as input, the operation returns the associated Veronese map.
gap> pg:=PG(2,5); ProjectiveSpace(2, 5) gap> vv:=VeroneseVariety(pg); Veronese Variety in ProjectiveSpace(5, 5) gap> Size(Points(vv))=Size(Points(pg)); true gap> vm:=VeroneseMap(vv); Veronese Map of <points of ProjectiveSpace(2, 5)> gap> r:=PolynomialRing(GF(5),3); GF(5)[x_1,x_2,x_3] gap> f:=r.1^2-r.2*r.3; x_1^2-x_2*x_3 gap> c:=AlgebraicVariety(pg,r,[f]); Projective Variety in ProjectiveSpace(2, 5) gap> pts:=List(Points(c)); [ <a point in ProjectiveSpace(2, 5)>, <a point in ProjectiveSpace(2, 5)>, <a point in ProjectiveSpace(2, 5)>, <a point in ProjectiveSpace(2, 5)>, <a point in ProjectiveSpace(2, 5)>, <a point in ProjectiveSpace(2, 5)> ] gap> Dimension(Span(ImagesSet(vm,pts))); 4
‣ Source( vm ) | ( operation ) |
Returns: the source of a Veronese map
The argument vm is a VeroneseMap (11.7-3). This operation returns the pointset of the projective space that was used to construct the VeroneseMap (11.7-3).
A Grassmann variety in FinInG is a projective algebraic variety in a projective space over a finite field. The set of points that lie on this variety is the image of the Grassmann map.
‣ GrassmannVariety( k, pg ) | ( operation ) |
‣ GrassmannVariety( subspaces ) | ( operation ) |
‣ GrassmannVariety( k, n, q ) | ( operation ) |
Returns: a Grassmann variety
The argument pg is a projective space defined over a finite field, say PG(n,q), and argument k is an integer (k at least 1 and at most n-2) and denotes the projective dimension determining the Grassmann Variety. The operation also takes as input the set subspaces of subspaces of a projective space, or the dimension k, the dimension n and the size q of the finite field (k at least 1 and at most n-2). The operation returns a projective algebraic variety known as the Grassmann variety.
‣ PointsOfGrassmannVariety( gv ) | ( operation ) |
‣ Points( gv ) | ( operation ) |
Returns: the points of a Grassmann variety
The argument gv is a Grassmann variety. This operation returns a set of points of the AmbientSpace (11.1-3) of the Grassmann variety. This set of points corresponds to the image of the GrassmannMap (11.8-3).
‣ GrassmannMap( k, pg ) | ( operation ) |
‣ GrassmannMap( subspaces ) | ( operation ) |
‣ GrassmannMap( k, n, q ) | ( operation ) |
‣ GrassmannMap( gv ) | ( operation ) |
Returns: a geometry map
The argument pg is a projective space defined over a finite field, say PG(n,q), and argument k is an integer (k at least 1 and at most n-2), and denotes the projective dimension determining the Grassmann Variety. The operation also takes as input the set subspaces of subspaces of a projective space, or the dimension k, the dimension n and the size q of the finite field (k at least 1 and at most n-2). The operation returns a function with domain the set of subspaces of dimension k in the n-dimensional projective space over GF(q), and image the set of points of the Grassmann variety (PointsOfGrassmannVariety (11.8-2)). When a Grassmann variety gv is given as input, the operation returns the associated Grassmann map.
‣ Source( gm ) | ( operation ) |
Returns: the source of a Grassmann map
The argument gm is a GrassmannMap (11.8-3). This operation returns the set of subspaces of the projective space that was used to construct the GrassmannMap (11.8-3).
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