AN INVARIANT FOR HYPERSURFACES IN ... - Semantic Scholar
AN INVARIANT FOR HYPERSURFACES IN PRIME CHARACTERISTIC DAVID G. GLYNN∗ Abstract. A hypersurface of order (n + 1)(ph − 1) in projective space of dimension n of prime characteristic p has an invariant monomial. This implies that a hypersurface of order (n+1)(ph −1)−1 determines an invariant point. A hypersurface of order d < n + 1 in a projective space of dimension n of characteristic two has an invariant set of subspaces of dimension d − 1 determined by one linear condition on the Grassmann coordinates of the dual subspaces. Key words. Projective space, prime, invariant, hypersurface, nucleus, linear complex, geometric code AMS subject classifications. 14L24, 51A05, 11C20, 11S05, 15A15, 15A72, 94B27
1. Introduction. Projective space PG(n, F ) of dimension n over a field F of prime characteristic p has points which have homogeneous coordinates (x0 , . . . , xn ) in the vector space F n+1 ; see [3, 4]. Let h ∈ Z+ . A hypersurface S of order αh := (n + 1)(ph − 1) is a set of points of PG(n, F ) satisfying a single homogeneous equation f (x0 , . . . , xn ) = 0 of degree αh . h There are n+α coefficients βm of S, when written as the sum of monomials f = n Σm βm m. In §2 we show that one of these coefficients is an invariant g(S). It follows that if A is any non-singular matrix transformation of PG(n, F ) in PGL(n + 1, F ), h then g(AS) = det(A)p −1 g(S). In particular g(S) = 0 if and only if g(AS) = 0. In §3 we consider hypersurfaces T of one less order. These hypersurfaces have an invariant point N (T ), the “nucleus” of T . This generalises the construction of the nucleus of a conic in a plane of characteristic two; see [4]. Then a result about hypersurfaces of order d < n + 1 in PG(n, F ) of characteristic two is given. 2. The Invariant. Here we show that the determinant-like invariant Xph of sets of points and matrices given in [1] implies an invariant for hypersurfaces of a particular order. Theorem 2.1. A hypersurface S : f = 0 of order αh = (n + 1)(ph − 1) in projective space of dimension n of prime characteristic p has an invariant Xph (S) h
h
that is the coefficient of xp0 −1 · · · xpn −1 in f . Proof. Write f as a sum Σm βm m of monomials m of degree (n + 1)(ph − 1), with βm ∈ F . Now m is the product of αh xi ’s and so m = 0 is the union M of αh hyperplanes of PG(n, F ). Dualising M to obtain M d we see that this is a multiset h h of αh points of PG(n, F ). In [1] it is shown that the coefficient of xp0 −1 · · · xpn −1 in h h m is the invariant Xph (M d ). Thus the coefficient of xp0 −1 · · · xpn −1 in S, equal to Σm βm Xph (M d ), is an invariant g. Xph (S) = 0, for F = GF(ph ), is equivalent to f summing to zero over all points in P G(n, F ); see [2]. This is one of the basic results that leads to the classification of the Reed-Muller geometric codes by polynomial functions. 3. Several Applications of the Invariant. We shall show that a general hypersurface T of one less order has an invariant point N (T ), which we call the “nucleus” of T . We now identify the hypersurface with its defining polynomial T = 0. ∗ CSEM, Flinders University of South Australia, P.O. Box 2100, Adelaide, South Australia 5001, Australia ([email protected], [email protected]).
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Theorem 3.1. A hypersurface T of order αh − 1 in PG(n, F ) has an invariant h h point (p0 , . . . , pn ), where pi is the coefficient of xp0 −1 · · · xpn −1 /xi in T . (If pi = 0 for all i, then the invariant point vanishes.) Proof. Consider the union T.u of T with a general hyperplane u : Σni=0 ui xi = 0 of PG(n, F ). Then Xph (T · u) = 0 is an invariant condition from Theorem 2.1 and it is a linear equation in the ui ’s. The coefficient of ui in this condition is the coefficient h h pi of xp0 −1 · · · xpn −1 /xi in T . Thus Xph (T · u) = 0 if and only if u lies on the point (p0 , . . . , pn ), which we could call the “nucleus” N (T ) of T . For example this gives the nucleus of a plane conic in characteristic two, and the extension in length of the corresponding Reed-Solomon MDS code. Another example is that in a projective plane of characteristic three a curve of order 3(3h − 1) − 1 has a nucleus. Also, in projective space of dimension three of characteristic p a surface of order 4(ph − 1) − 1 has a nucleus. It is not clear if this kind of nucleus can always be obtained by tangential properties of the hypersurface, although in several cases such as the conic case above it is. Theorem 3.2. A hypersurface K of order d < n + 1 in PG(n, F ), where F has characteristic two, has an invariant linear complex of subspaces of dual dimension n − d. Proof. Consider the equation X2 (K · v1 · · · vn+1−d ) = 0, where the vi are variable hyperplanes (linear functions in the xj ’s). If V is the (n + 1) × (n + 1 − d) matrix over F with columns corresponding to the homogeneous coordinates of the vi ’s, then each subset of (n + 1 − d) rows of V corresponds to a subdeterminant of V , which is the coefficient of a unique monomial of degree n + 1 − d in distinct xi ’s of v1 · · · vn+1−d . It is also one of the Grassmannian coordinates of the subspace spV := colspace(V ) = hv1 , . . . , vn+1−d i of dimension n − d in the dual space of PG(n, F ). The coefficient of the diagonal monomial x0 · · · xn in K ·v1 · · · vn+1−d is the invariant X2 and this is zero if and only if the Grassmannian coordinates of spV satisfy a certain linear equation, in which the coefficient of the coordinate related to the subset {r1 , . . . , rn+1−d } of {1, . . . , n+1} is the coefficient of xs1 . . . xsd in K, where {r1 , . . . , rn+1−d , s1 , . . . , sd } = {1, . . . , n + 1}. Note that the hyperplanes generating an invariant dual subspace of dimension n − d must intersect also in an invariant subspace (of points) of dimension d − 1. Thus an invariant set of subspaces of dimension d − 1 is determined from the hypersurface of order d in PG(n, F ) of characteristic two. A special case of this is the well-known property of quadrics in three-dimensional projective space of characteristic two: that their tangent lines form a linear complex, being the totally isotropic lines with respect to a symplectic polarity. REFERENCES [1] D. G. Glynn, An invariant for matrices and sets of points in prime characteristic, Des. Codes Cryptogr., 58 (2011), pp. 155–172. DOI: 10.1007/s10623-010-9392-x. [2] D. G. Glynn and J. W. P. Hirschfeld, On the classification of geometric codes by polynomial functions, Des. Codes Cryptogr., 6 (1995), pp. 189–204. [3] J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991. ´ ros, and F. Torres, Algebraic Curves over a Finite Field, [4] J. W. P. Hirschfeld, G. Korchma Princeton University Press, 2008.
1. Introduction. Projective space PG(n, F ) of dimension n over a field F of prime characteristic p has points which have homogeneous coordinates (x0 , . . . , xn ) in the vector space F n+1 ; see [3, 4]. Let h ∈ Z+ . A hypersurface S of order αh := (n + 1)(ph − 1) is a set of points of PG(n, F ) satisfying a single homogeneous equation f (x0 , . . . , xn ) = 0 of degree αh . h There are n+α coefficients βm of S, when written as the sum of monomials f = n Σm βm m. In §2 we show that one of these coefficients is an invariant g(S). It follows that if A is any non-singular matrix transformation of PG(n, F ) in PGL(n + 1, F ), h then g(AS) = det(A)p −1 g(S). In particular g(S) = 0 if and only if g(AS) = 0. In §3 we consider hypersurfaces T of one less order. These hypersurfaces have an invariant point N (T ), the “nucleus” of T . This generalises the construction of the nucleus of a conic in a plane of characteristic two; see [4]. Then a result about hypersurfaces of order d < n + 1 in PG(n, F ) of characteristic two is given. 2. The Invariant. Here we show that the determinant-like invariant Xph of sets of points and matrices given in [1] implies an invariant for hypersurfaces of a particular order. Theorem 2.1. A hypersurface S : f = 0 of order αh = (n + 1)(ph − 1) in projective space of dimension n of prime characteristic p has an invariant Xph (S) h
h
that is the coefficient of xp0 −1 · · · xpn −1 in f . Proof. Write f as a sum Σm βm m of monomials m of degree (n + 1)(ph − 1), with βm ∈ F . Now m is the product of αh xi ’s and so m = 0 is the union M of αh hyperplanes of PG(n, F ). Dualising M to obtain M d we see that this is a multiset h h of αh points of PG(n, F ). In [1] it is shown that the coefficient of xp0 −1 · · · xpn −1 in h h m is the invariant Xph (M d ). Thus the coefficient of xp0 −1 · · · xpn −1 in S, equal to Σm βm Xph (M d ), is an invariant g. Xph (S) = 0, for F = GF(ph ), is equivalent to f summing to zero over all points in P G(n, F ); see [2]. This is one of the basic results that leads to the classification of the Reed-Muller geometric codes by polynomial functions. 3. Several Applications of the Invariant. We shall show that a general hypersurface T of one less order has an invariant point N (T ), which we call the “nucleus” of T . We now identify the hypersurface with its defining polynomial T = 0. ∗ CSEM, Flinders University of South Australia, P.O. Box 2100, Adelaide, South Australia 5001, Australia ([email protected], [email protected]).
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Theorem 3.1. A hypersurface T of order αh − 1 in PG(n, F ) has an invariant h h point (p0 , . . . , pn ), where pi is the coefficient of xp0 −1 · · · xpn −1 /xi in T . (If pi = 0 for all i, then the invariant point vanishes.) Proof. Consider the union T.u of T with a general hyperplane u : Σni=0 ui xi = 0 of PG(n, F ). Then Xph (T · u) = 0 is an invariant condition from Theorem 2.1 and it is a linear equation in the ui ’s. The coefficient of ui in this condition is the coefficient h h pi of xp0 −1 · · · xpn −1 /xi in T . Thus Xph (T · u) = 0 if and only if u lies on the point (p0 , . . . , pn ), which we could call the “nucleus” N (T ) of T . For example this gives the nucleus of a plane conic in characteristic two, and the extension in length of the corresponding Reed-Solomon MDS code. Another example is that in a projective plane of characteristic three a curve of order 3(3h − 1) − 1 has a nucleus. Also, in projective space of dimension three of characteristic p a surface of order 4(ph − 1) − 1 has a nucleus. It is not clear if this kind of nucleus can always be obtained by tangential properties of the hypersurface, although in several cases such as the conic case above it is. Theorem 3.2. A hypersurface K of order d < n + 1 in PG(n, F ), where F has characteristic two, has an invariant linear complex of subspaces of dual dimension n − d. Proof. Consider the equation X2 (K · v1 · · · vn+1−d ) = 0, where the vi are variable hyperplanes (linear functions in the xj ’s). If V is the (n + 1) × (n + 1 − d) matrix over F with columns corresponding to the homogeneous coordinates of the vi ’s, then each subset of (n + 1 − d) rows of V corresponds to a subdeterminant of V , which is the coefficient of a unique monomial of degree n + 1 − d in distinct xi ’s of v1 · · · vn+1−d . It is also one of the Grassmannian coordinates of the subspace spV := colspace(V ) = hv1 , . . . , vn+1−d i of dimension n − d in the dual space of PG(n, F ). The coefficient of the diagonal monomial x0 · · · xn in K ·v1 · · · vn+1−d is the invariant X2 and this is zero if and only if the Grassmannian coordinates of spV satisfy a certain linear equation, in which the coefficient of the coordinate related to the subset {r1 , . . . , rn+1−d } of {1, . . . , n+1} is the coefficient of xs1 . . . xsd in K, where {r1 , . . . , rn+1−d , s1 , . . . , sd } = {1, . . . , n + 1}. Note that the hyperplanes generating an invariant dual subspace of dimension n − d must intersect also in an invariant subspace (of points) of dimension d − 1. Thus an invariant set of subspaces of dimension d − 1 is determined from the hypersurface of order d in PG(n, F ) of characteristic two. A special case of this is the well-known property of quadrics in three-dimensional projective space of characteristic two: that their tangent lines form a linear complex, being the totally isotropic lines with respect to a symplectic polarity. REFERENCES [1] D. G. Glynn, An invariant for matrices and sets of points in prime characteristic, Des. Codes Cryptogr., 58 (2011), pp. 155–172. DOI: 10.1007/s10623-010-9392-x. [2] D. G. Glynn and J. W. P. Hirschfeld, On the classification of geometric codes by polynomial functions, Des. Codes Cryptogr., 6 (1995), pp. 189–204. [3] J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991. ´ ros, and F. Torres, Algebraic Curves over a Finite Field, [4] J. W. P. Hirschfeld, G. Korchma Princeton University Press, 2008.
