Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe - PowerPoint PPT Presentation

Definition of equivalence of linear sets L U and L V F q -linear sets of Λ = PG ( W , F q n ) = PG ( r − 1 , q n ) L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ ∈ P Γ L ( r , q n ) s.t. L Φ U = L V Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 6 / 23

Definition of equivalence of linear sets L U and L V F q -linear sets of Λ = PG ( W , F q n ) = PG ( r − 1 , q n ) L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ ∈ P Γ L ( r , q n ) s.t. L Φ U = L V U = V f f ∈ Γ L ( r , q n ) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 6 / 23

Definition of equivalence of linear sets L U and L V F q -linear sets of Λ = PG ( W , F q n ) = PG ( r − 1 , q n ) L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ ∈ P Γ L ( r , q n ) s.t. L Φ U = L V f ∈ Γ L ( r , q n ) ⇒ L Φ f U = V f U = L U f = L V Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 6 / 23

Definition of equivalence of linear sets L U and L V F q -linear sets of Λ = PG ( W , F q n ) = PG ( r − 1 , q n ) L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ ∈ P Γ L ( r , q n ) s.t. L Φ U = L V f ∈ Γ L ( r , q n ) ⇒ L Φ f U = V f U = L U f = L V The converse does not hold Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 6 / 23

Definition of equivalence of linear sets L U and L V F q -linear sets of Λ = PG ( W , F q n ) = PG ( r − 1 , q n ) L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ ∈ P Γ L ( r , q n ) s.t. L Φ U = L V f ∈ Γ L ( r , q n ) ⇒ L Φ f U = V f U = L U f = L V The converse does not hold Example F q -vector subspaces of W = V ( r , q n ) of rank k ≥ rn − n + 1 determine the whole projective space but there is no semilinear map between two F q -subspaces with different rank Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 6 / 23

Definition of equivalence of linear sets L U and L V F q -linear sets of Λ = PG ( W , F q n ) = PG ( r − 1 , q n ) L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ ∈ P Γ L ( r , q n ) s.t. L Φ U = L V f ∈ Γ L ( r , q n ) ⇒ L Φ f U = V f U = L U f = L V The converse does not hold Example F q -vector subspaces of W = V ( 2 , q n ) of rank k ≥ 2 n − n + 1 determine the whole projective space but there is no semilinear map between two F q -subspaces with different rank Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 6 / 23

Equivalence issue linear sets of rank n in PG ( 1 , q n ) L U an F q -linear set of rank n in PG ( 1 , q n ) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 7 / 23

Equivalence issue linear sets of rank n in PG ( 1 , q n ) L U an F q -linear set of rank n in PG ( 1 , q n ) L V is equivalent to L U ⇒ Φ f ∈ P Γ L ( 2 , q n ) s.t. L Φ f V = L V f = L U Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 7 / 23

Equivalence issue linear sets of rank n in PG ( 1 , q n ) L U an F q -linear set of rank n in PG ( 1 , q n ) L V is equivalent to L U ⇒ Φ f ∈ P Γ L ( 2 , q n ) , f ∈ Γ L ( 2 , q n ) , s.t. L Φ f V = L V f = L U Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 7 / 23

Equivalence issue linear sets of rank n in PG ( 1 , q n ) L U an F q -linear set of rank n in PG ( 1 , q n ) L V is equivalent to L U ⇒ Φ f ∈ P Γ L ( 2 , q n ) , f ∈ Γ L ( 2 , q n ) , s.t. L Φ f V = L V f = L U FIRST STEP: Determine all F q -subspaces defining L U Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 7 / 23

Equivalence issue linear sets of rank n in PG ( 1 , q n ) L U an F q -linear set of rank n in PG ( 1 , q n ) L V is equivalent to L U ⇒ Φ f ∈ P Γ L ( 2 , q n ) , f ∈ Γ L ( 2 , q n ) , s.t. L Φ f V = L V f = L U FIRST STEP: Determine all F q -subspaces defining L U Question Is it possible to have an F q -subspace of rank different from n defining L U ? Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 7 / 23

Equivalence between F q -linear sets of PG ( 1 , q n ) of rank n Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 8 / 23

Equivalence between F q -linear sets of PG ( 1 , q n ) of rank n Theorem ( Ball, Blokhuis, Brouwer, Storme, Sz˝ onyi, 1999 - Ball, 2003) Let f be a function from F q to F q , q = p h , and let N be the number of directions determined by f. Let s = p e be maximal such that any line with a direction determined by f that is incident with a point of the graph of f is incident with a multiple of s points of the graph of f. Then one of the following holds. s = 1 and ( q + 3 ) / 2 ≤ N ≤ q + 1 , 1 e | h, q / s + 1 ≤ N ≤ ( q − 1 ) / ( s − 1 ) , 2 s = q and N = 1 . 3 Moreover if s > 2 , then the graph of f is F s -linear. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 8 / 23

Equivalence between F q -linear sets of PG ( 1 , q n ) of rank n F q t is the maximum field of linearity of L U if t | n and L U is an F q t -linear set Theorem (B. Csajbók, G.M., O. Polverino) Let L U be an F q -linear set of PG ( W , F q n ) = PG ( 1 , q n ) of rank n. The maximum field of linearity of L U is F q d , where d = min { dim q ( U ∩ � u � q n ): u ∈ U \ { 0 }} . If the maximum field of linearity of L U is F q , then the rank of L U as an F q -linear set is uniquely defined, i.e. for each F q -subspace V of W if L U = L V , then dim q ( V ) = n. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 9 / 23

Equivalence between F q -linear sets of PG ( 1 , q n ) of rank n F q t is the maximum field of linearity of L U if t | n and L U is an F q t -linear set Theorem (B. Csajbók, G.M., O. Polverino) Let L U be an F q -linear set of PG ( W , F q n ) = PG ( 1 , q n ) of rank n. The maximum field of linearity of L U is F q d , where d = min { dim q ( U ∩ � u � q n ): u ∈ U \ { 0 }} . If the maximum field of linearity of L U is F q , then the rank of L U as an F q -linear set is uniquely defined, i.e. for each F q -subspace V of W if L U = L V , then dim q ( V ) = n. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 9 / 23

Equivalence of linear sets in PG ( 1 , q n ) of rank n L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ f ∈ P Γ L ( 2 , q n ) s.t. L Φ f U = L U f = L V Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 10 / 23

Equivalence of linear sets in PG ( 1 , q n ) of rank n L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ f ∈ P Γ L ( 2 , q n ) s.t. L Φ f U = L U f = L V FIRST STEP: Determine all F q -subspaces defining L U (which have all rank n ) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 10 / 23

Equivalence of linear sets in PG ( 1 , q n ) of rank n L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ f ∈ P Γ L ( 2 , q n ) s.t. L Φ f U = L U f = L V FIRST STEP: Determine all F q -subspaces defining L U (which have all rank n ) SECOND STEP: Study the action on these F q -subspaces of Γ L ( 2 , q n ) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 10 / 23

Equivalence of linear sets in PG ( 1 , q n ) of rank n L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ f ∈ P Γ L ( 2 , q n ) s.t. L Φ f U = L U f = L V FIRST STEP: Determine all F q -subspaces defining L U (which have all rank n ) SECOND STEP: Study the action on these F q -subspaces of Γ L ( 2 , q n ) Definition Let L U be an F q -linear set of PG ( W , F q n ) = PG ( 1 , q n ) of rank n with maximum field of linearity F q . The Γ L -class of L U is the number of the Γ L ( 2 , q n ) -orbits determined by the F q -subspaces defining L U . Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 10 / 23

Equivalence of linear sets in PG ( 1 , q n ) of rank n L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ f ∈ P Γ L ( 2 , q n ) s.t. L Φ f U = L U f = L V FIRST STEP: Determine all F q -subspaces defining L U (which have all rank n ) SECOND STEP: Study the action on these F q -subspaces of Γ L ( 2 , q n ) Definition Let L U be an F q -linear set of PG ( W , F q n ) = PG ( 1 , q n ) of rank n with maximum field of linearity F q . The Γ L -class of L U is the number of the Γ L ( 2 , q n ) -orbits determined by the F q -subspaces defining L U . The Γ L -class of a linear set is a P Γ L -invariant Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 10 / 23

Equivalence of linear sets in PG ( 1 , q n ) of rank n L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ f ∈ P Γ L ( 2 , q n ) s.t. L Φ f U = L U f = L V FIRST STEP: Determine all F q -subspaces defining L U (which have all rank n ) SECOND STEP: Study the action on these F q -subspaces of Γ L ( 2 , q n ) Definition Let L U be an F q -linear set of PG ( W , F q n ) = PG ( 1 , q n ) of rank n with maximum field of linearity F q . The Γ L -class of L U is the number of the Γ L ( 2 , q n ) -orbits determined by the F q -subspaces defining L U . If the Γ L -class is 1, then L U is said to be simple Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 10 / 23

Equivalence of linear sets in PG ( 1 , q n ) of rank n L U and L V are P Γ L -equivalent (or simply equivalent ) if there is an element Φ f ∈ P Γ L ( 2 , q n ) s.t. L Φ f U = L U f = L V FIRST STEP: Determine all F q -subspaces defining L U (which have all rank n ) SECOND STEP: Study the action on these F q -subspaces of Γ L ( 2 , q n ) Definition Let L U be an F q -linear set of PG ( W , F q n ) = PG ( 1 , q n ) of rank n with maximum field of linearity F q . The Γ L -class of L U is the number of the Γ L ( 2 , q n ) -orbits determined by the F q -subspaces defining L U . If the Γ L -class is 1, then L U is said to be simple Simple linear sets have been also studied by Csajboók-Zanella and Van de Voorde Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 10 / 23

Simple linear sets Definition An F q -linear set L of PG ( r − 1 , q n ) = PG ( W , F q n ) of rank k with maximum field of linearity F q is called simple if all the F q -subspaces of W of dimension k defining L are in the same orbit of Γ L ( r , q n ) . Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 11 / 23

Simple linear sets Definition An F q -linear set L of PG ( r − 1 , q n ) = PG ( W , F q n ) of rank k with maximum field of linearity F q is called simple if all the F q -subspaces of W of dimension k defining L are in the same orbit of Γ L ( r , q n ) . Example Subgeometries (trivial). Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 11 / 23

Simple linear sets Definition An F q -linear set L of PG ( r − 1 , q n ) = PG ( W , F q n ) of rank k with maximum field of linearity F q is called simple if all the F q -subspaces of W of dimension k defining L are in the same orbit of Γ L ( r , q n ) . Example Subgeometries (trivial). Remark Let L U and L V be two F q -linear sets of PG ( r − 1 , q n ) of rank k. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 11 / 23

Simple linear sets Definition An F q -linear set L of PG ( r − 1 , q n ) = PG ( W , F q n ) of rank k with maximum field of linearity F q is called simple if all the F q -subspaces of W of dimension k defining L are in the same orbit of Γ L ( r , q n ) . Example Subgeometries (trivial). Remark Let L U and L V be two F q -linear sets of PG ( r − 1 , q n ) of rank k. If L U is simple, then L V is P Γ L -equivalent to L U iff U and V are Γ L ( r , q n ) -equivalent Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 11 / 23

Simple linear sets Definition An F q -linear set L of PG ( r − 1 , q n ) = PG ( W , F q n ) of rank k with maximum field of linearity F q is called simple if all the F q -subspaces of W of dimension k defining L are in the same orbit of Γ L ( r , q n ) . Example Subgeometries (trivial). Remark Let L U and L V be two F q -linear sets of PG ( r − 1 , q n ) of rank k. If L U is simple, then L V is P Γ L -equivalent to L U iff U and V are Γ L ( r , q n ) -equivalent Example (Bonoli-Polverino, 2005) F q -linear sets of PG ( 2 , q n ) of rank n + 1 with ( q + 1 ) -secants are simple. This allowed a complete classification of F q -linear blocking sets in PG ( 2 , q 4 ) . Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 11 / 23

Non-simple F q -linear sets of PG ( 1 , q n ) of rank n Example (Csajbók-Zanella, 2016) Linear sets of pseudoregulus type of PG ( 1 , q n ) L U = {� ( x , x q s ) � : x ∈ F ∗ q n } , gcd ( s , n ) = 1 are non-simple for n ≥ 5, n � = 6. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 12 / 23

Non-simple F q -linear sets of PG ( 1 , q n ) of rank n Example (Csajbók-Zanella, 2016) Linear sets of pseudoregulus type of PG ( 1 , q n ) L U = {� ( x , x q s ) � : x ∈ F ∗ q n } , gcd ( s , n ) = 1 are non-simple for n ≥ 5, n � = 6. It is not hard to find non-simple linear sets! Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 12 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by β : W × W → F q n non-degenerate alternating form Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by β : W × W → F q n non-degenerate alternating form ↓ Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by β : W × W → F q n non-degenerate alternating form ↓ L τ U := L U ⊥ dual linear set Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by β : W × W → F q n non-degenerate alternating form ↓ L τ U := L U ⊥ dual linear set U ⊥ orthogonal complement of U wrt Tr q n / q ◦ β : W × W → F q Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by β : W × W → F q n non-degenerate alternating form ↓ L τ U := L U ⊥ dual linear set (rank n ) U ⊥ orthogonal complement of U wrt Tr q n / q ◦ β : W × W → F q Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by β : W × W → F q n non-degenerate alternating form ↓ L τ U := L U ⊥ dual linear set (rank n ) U ⊥ orthogonal complement of U wrt Tr q n / q ◦ β : W × W → F q Up to projective equivalence such a linear set does not depend on τ Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by β : W × W → F q n non-degenerate alternating form ↓ L τ U := L U ⊥ dual linear set (rank n ) U ⊥ orthogonal complement of U wrt Tr q n / q ◦ β : W × W → F q If τ is symplectic Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set L U F q -linear set of rank n of PG ( 1 , q n ) τ polarity of PG ( 1 , q n ) = PG ( W , F q n ) induced by β : W × W → F q n non-degenerate alternating form ↓ L τ U := L U ⊥ dual linear set (rank n ) U ⊥ orthogonal complement of U wrt Tr q n / q ◦ β : W × W → F q If τ is symplectic then L U = L τ U = L U ⊥ Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 13 / 23

Dual of a linear set In practice: L U , U := U f = { ( x , f ( x )): x ∈ F q n } , i = 0 a i x q i , a i ∈ F q n for some q -polynomial f ( x ) = � n − 1 Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 14 / 23

Dual of a linear set In practice: L U , U := U f = { ( x , f ( x )): x ∈ F q n } , i = 0 a i x q i , a i ∈ F q n for some q -polynomial f ( x ) = � n − 1 τ symplectic polarity of PG ( 1 , q n ) induced by β (( x , y ) , ( u , v )) := xv − uy Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 14 / 23

Dual of a linear set In practice: L U , U := U f = { ( x , f ( x )): x ∈ F q n } , i = 0 a i x q i , a i ∈ F q n for some q -polynomial f ( x ) = � n − 1 τ symplectic polarity of PG ( 1 , q n ) induced by β (( x , y ) , ( u , v )) := xv − uy f = { ( x , ˆ U ⊥ = U ˆ f ( x )): x ∈ F q n } , f x q n − i is the adjoint map of f wrt the bilinear form � x , y � = Tr ( xy ) i = 0 a q n − i where ˆ f ( x ) := � n − 1 i Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 14 / 23

Dual of a linear set In practice: L U , U := U f = { ( x , f ( x )): x ∈ F q n } , i = 0 a i x q i , a i ∈ F q n for some q -polynomial f ( x ) = � n − 1 τ symplectic polarity of PG ( 1 , q n ) induced by β (( x , y ) , ( u , v )) := xv − uy f = { ( x , ˆ U ⊥ = U ˆ f ( x )): x ∈ F q n } , f x q n − i is the adjoint map of f wrt the bilinear form � x , y � = Tr ( xy ) i = 0 a q n − i where ˆ f ( x ) := � n − 1 i f are in different Γ L ( 2 , q n ) -orbits In general, U f and U ˆ Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 14 / 23

Dual of a linear set In practice: L U , U := U f = { ( x , f ( x )): x ∈ F q n } , i = 0 a i x q i , a i ∈ F q n for some q -polynomial f ( x ) = � n − 1 τ symplectic polarity of PG ( 1 , q n ) induced by β (( x , y ) , ( u , v )) := xv − uy f = { ( x , ˆ U ⊥ = U ˆ f ( x )): x ∈ F q n } , f x q n − i is the adjoint map of f wrt the bilinear form � x , y � = Tr ( xy ) i = 0 a q n − i where ˆ f ( x ) := � n − 1 i f are in different Γ L ( 2 , q n ) -orbits In general, U f and U ˆ ↓ Hence, usually, the Γ L -class of L U is at least 2 Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 14 / 23

Dual of a linear set In practice: L U , U := U f = { ( x , f ( x )): x ∈ F q n } , i = 0 a i x q i , a i ∈ F q n for some q -polynomial f ( x ) = � n − 1 τ symplectic polarity of PG ( 1 , q n ) induced by β (( x , y ) , ( u , v )) := xv − uy f = { ( x , ˆ U ⊥ = U ˆ f ( x )): x ∈ F q n } , f x q n − i is the adjoint map of f wrt the bilinear form � x , y � = Tr ( xy ) i = 0 a q n − i where ˆ f ( x ) := � n − 1 i f are in different Γ L ( 2 , q n ) -orbits In general, U f and U ˆ ↓ Hence, usually, the Γ L -class of L U is at least 2, i.e. L U is non-simple Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 14 / 23

Non-simple linear sets of rank n in PG ( 1 , q n ) Example (Csajbók-Zanella, 2016) F q -linear sets of PG ( 1 , q n ) of psudoregulus type L U = {� ( x , x q s ) � : x ∈ F ∗ q n } , gcd ( s , n ) = 1 The Γ L -class of L U is ϕ ( n ) / 2. Hence, for n ≥ 5 and n = 6, L U is not simple. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 15 / 23

Non-simple linear sets of rank n in PG ( 1 , q n ) Example (Csajbók-Zanella, 2016) F q -linear sets of PG ( 1 , q n ) of psudoregulus type L U = {� ( x , x q s ) � : x ∈ F ∗ q n } , gcd ( s , n ) = 1 The Γ L -class of L U is ϕ ( n ) / 2. Hence, for n ≥ 5 and n = 6, L U is not simple. Proposition (Csajbók-G.M.-Polverino) The F q -linear sets of PG ( 1 , q n ) introduced by Lunardon-Polverino (2001) L U = {� ( x , δ x q + x q n − 1 ) � : x ∈ F ∗ q n } , n > 3 , q ≥ 3 Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 15 / 23

Non-simple linear sets of rank n in PG ( 1 , q n ) Example (Csajbók-Zanella, 2016) F q -linear sets of PG ( 1 , q n ) of psudoregulus type L U = {� ( x , x q s ) � : x ∈ F ∗ q n } , gcd ( s , n ) = 1 The Γ L -class of L U is ϕ ( n ) / 2. Hence, for n ≥ 5 and n = 6, L U is not simple. Proposition (Csajbók-G.M.-Polverino) The F q -linear sets of PG ( 1 , q n ) introduced by Lunardon-Polverino (2001) L U = {� ( x , δ x q + x q n − 1 ) � : x ∈ F ∗ q n } , n > 3 , q ≥ 3 are not simple for n > 4 , q > 4 and δ a generator of F ∗ q n . Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 15 / 23

Non-simple linear sets of rank n in PG ( 1 , q n ) Example (Csajbók-Zanella, 2016) F q -linear sets of PG ( 1 , q n ) of psudoregulus type L U = {� ( x , x q s ) � : x ∈ F ∗ q n } , gcd ( s , n ) = 1 The Γ L -class of L U is ϕ ( n ) / 2. Hence, for n ≥ 5 and n = 6, L U is not simple. Proposition (Csajbók-G.M.-Polverino) The F q -linear sets of PG ( 1 , q n ) introduced by Lunardon-Polverino (2001) L U = {� ( x , δ x q + x q n − 1 ) � : x ∈ F ∗ q n } , n > 3 , q ≥ 3 are not simple for n > 4 , q > 4 and δ a generator of F ∗ q n . Other examples in PG ( 1 , q n ) , n ∈ { 6 , 8 } (Csajbók-G.M.-Polverino-Zanella, Csajbók-G.M.-Zullo) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 15 / 23

Non-simple linear sets of rank n in PG ( 1 , q n ) Example (Csajbók-Zanella, 2016) F q -linear sets of PG ( 1 , q n ) of psudoregulus type L U = {� ( x , x q s ) � : x ∈ F ∗ q n } , gcd ( s , n ) = 1 The Γ L -class of L U is ϕ ( n ) / 2. Hence, for n ≥ 5 and n = 6, L U is not simple. Proposition (Csajbók-G.M.-Polverino) The F q -linear sets of PG ( 1 , q n ) introduced by Lunardon-Polverino (2001) L U = {� ( x , δ x q + x q n − 1 ) � : x ∈ F ∗ q n } , n > 3 , q ≥ 3 are not simple for n > 4 , q > 4 and δ a generator of F ∗ q n . Other examples in PG ( 1 , q n ) , n ∈ { 6 , 8 } (Csajbók-G.M.-Polverino-Zanella, Csajbók-G.M.-Zullo) : Ferdinando ′ s talk ! Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 15 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Question Is it possible to find a simple F q -linear set of rank n in PG ( 1 , q n ) for each n? Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 16 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 17 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Lemma i = 0 a i x q i and g ( x ) = � n − 1 i = 0 b i x q i be two q-polynomials over F q n , such that Let f ( x ) = � n − 1 L f = L g , i.e. � f ( x ) � g ( x ) � � : x ∈ F ∗ : x ∈ F ∗ = . q n q n x x Then a 0 = b 0 , (1) and for k = 1 , 2 , . . . , n − 1 it holds that a k a q k n − k = b k b q k n − k , (2) for k = 2 , 3 , . . . , n − 1 it holds that k − 1 a q k n − 1 a q k k − 1 b q k n − 1 b q k a 1 a q n − k + a k a q n − k + 1 = b 1 b q n − k + b k b q n − k + 1 . (3) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 17 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Lemma i = 0 a i x q i and g ( x ) = � n − 1 i = 0 b i x q i be two q-polynomials over F q n , such that Let f ( x ) = � n − 1 L f = L g , i.e. � f ( x ) � g ( x ) � � : x ∈ F ∗ : x ∈ F ∗ = . q n q n x x Then a 0 = b 0 , (1) and for k = 1 , 2 , . . . , n − 1 it holds that a k a q k n − k = b k b q k n − k , (2) for k = 2 , 3 , . . . , n − 1 it holds that k − 1 a q k n − 1 a q k k − 1 b q k n − 1 b q k a 1 a q n − k + a k a q n − k + 1 = b 1 b q n − k + b k b q n − k + 1 . (3) Theorem Let T = { ( x , Tr q n / q ( x )): x ∈ F q n } ⊂ PG ( 1 , q n ) = PG ( W , F q n ) . For each F q -subspace U of W it turns out L U = L T only if T = λ U for some λ ∈ F ∗ q n . Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 17 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Lemma i = 0 a i x q i and g ( x ) = � n − 1 i = 0 b i x q i be two q-polynomials over F q n , such that Let f ( x ) = � n − 1 L f = L g , i.e. � f ( x ) � g ( x ) � � : x ∈ F ∗ : x ∈ F ∗ = . q n q n x x Then a 0 = b 0 , (1) and for k = 1 , 2 , . . . , n − 1 it holds that a k a q k n − k = b k b q k n − k , (2) for k = 2 , 3 , . . . , n − 1 it holds that k − 1 a q k n − 1 a q k k − 1 b q k n − 1 b q k a 1 a q n − k + a k a q n − k + 1 = b 1 b q n − k + b k b q n − k + 1 . (3) Theorem Let T = { ( x , Tr q n / q ( x )): x ∈ F q n } ⊂ PG ( 1 , q n ) = PG ( W , F q n ) . For each F q -subspace U of W it turns out L U = L T only if T = λ U for some λ ∈ F ∗ q n . Hence, L T simple. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 17 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 18 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Summing up: Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 18 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Summing up: L T is simple for each n Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 18 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Summing up: L T is simple for each n For n > 4 there are non-simple linear sets (linear sets of Lunardon-Polverino and linear sets of pseudoregulus type) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 18 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Summing up: L T is simple for each n For n > 4 there are non-simple linear sets (linear sets of Lunardon-Polverino and linear sets of pseudoregulus type) n = 2 → Baer sublines (simple) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 18 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Summing up: L T is simple for each n For n > 4 there are non-simple linear sets (linear sets of Lunardon-Polverino and linear sets of pseudoregulus type) n = 2 → Baer sublines (simple) n = 3 → Pseudoregulus type (simple) Clubs (simple) Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 18 / 23

Simple F q -linear sets of PG ( 1 , q n ) of rank n Summing up: L T is simple for each n For n > 4 there are non-simple linear sets (linear sets of Lunardon-Polverino and linear sets of pseudoregulus type) n = 2 → Baer sublines (simple) n = 3 → Pseudoregulus type (simple) Clubs (simple) Question What happens for n = 4 ? Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 18 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Theorem Linear sets of rank 4 of PG ( 1 , q 4 ) , with maximum field of linearity F q , are simple. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Theorem Linear sets of rank 4 of PG ( 1 , q 4 ) , with maximum field of linearity F q , are simple. Sketch of Proof. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Theorem Linear sets of rank 4 of PG ( 1 , q 4 ) , with maximum field of linearity F q , are simple. Sketch of Proof. 1 Simplicity is P Γ L -invariant, so we can consider linear sets of type L f = L U f , i = 0 a i x q i U f = { ( x , f ( x )) : x ∈ F q 4 } , with f ( x ) = � 4 Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Theorem Linear sets of rank 4 of PG ( 1 , q 4 ) , with maximum field of linearity F q , are simple. Sketch of Proof. 1 Simplicity is P Γ L -invariant, so we can consider linear sets of type L f = L U f , i = 0 a i x q i U f = { ( x , f ( x )) : x ∈ F q 4 } , with f ( x ) = � 4 i = 0 b i x q i such that L f = L g . By technical lemma we have Let g ( x ) = � 4 2 3 , a q 2 + 1 = b q 2 + 1 a q 2 2 + a 2 a q + q 2 b q 2 2 + b 2 b q + q 2 a 0 = b 0 , a 1 a q 3 = b 1 b q , a q + 1 = b q + 1 2 2 1 3 1 3 Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Theorem Linear sets of rank 4 of PG ( 1 , q 4 ) , with maximum field of linearity F q , are simple. Sketch of Proof. 1 Simplicity is P Γ L -invariant, so we can consider linear sets of type L f = L U f , i = 0 a i x q i U f = { ( x , f ( x )) : x ∈ F q 4 } , with f ( x ) = � 4 i = 0 b i x q i such that L f = L g . By technical lemma we have Let g ( x ) = � 4 2 3 , a q 2 + 1 = b q 2 + 1 a q 2 2 + a 2 a q + q 2 b q 2 2 + b 2 b q + q 2 a 0 = b 0 , a 1 a q 3 = b 1 b q , a q + 1 = b q + 1 2 2 1 3 1 3 3 Also, for n = 4, we have � � N q n / q ( a 1 ) + N q n / q ( a 2 ) + N q n / q ( a 3 ) + a 1 + q 2 a q + q 3 + a q + q 3 a 1 + q 2 a 1 a q + q 2 a q 3 + Tr q 4 / q = 1 3 1 3 2 3 � � N q n / q ( b 1 ) + N q n / q ( b 2 ) + N q n / q ( b 3 ) + b 1 + q 2 b q + q 3 + b q + q 3 b 1 + q 2 b 1 b q + q 2 b q 3 + Tr q 4 / q 1 3 1 3 2 3 Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Theorem Linear sets of rank 4 of PG ( 1 , q 4 ) , with maximum field of linearity F q , are simple. Sketch of Proof. 1 Simplicity is P Γ L -invariant, so we can consider linear sets of type L f = L U f , i = 0 a i x q i U f = { ( x , f ( x )) : x ∈ F q 4 } , with f ( x ) = � 4 i = 0 b i x q i such that L f = L g . Then there exists λ ∈ F ∗ Let g ( x ) = � 4 q 4 such that 4 U g = λ U f or U g = λ U ˆ f . Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Theorem Linear sets of rank 4 of PG ( 1 , q 4 ) , with maximum field of linearity F q , are simple. Sketch of Proof. 1 Simplicity is P Γ L -invariant, so we can consider linear sets of type L f = L U f , i = 0 a i x q i U f = { ( x , f ( x )) : x ∈ F q 4 } , with f ( x ) = � 4 i = 0 b i x q i such that L f = L g . Then there exists λ ∈ F ∗ Let g ( x ) = � 4 q 4 such that 4 U g = λ U f or U g = λ U ˆ f . Hence the Γ L -class of L f is at most 2. Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 Theorem Linear sets of rank 4 of PG ( 1 , q 4 ) , with maximum field of linearity F q , are simple. Sketch of Proof. 1 Simplicity is P Γ L -invariant, so we can consider linear sets of type L f = L U f , i = 0 a i x q i U f = { ( x , f ( x )) : x ∈ F q 4 } , with f ( x ) = � 4 i = 0 b i x q i such that L f = L g . Then there exists λ ∈ F ∗ Let g ( x ) = � 4 q 4 such that 4 U g = λ U f or U g = λ U ˆ f . Hence the Γ L -class of L f is at most 2. f are in the same Γ L ( 2 , q 4 ) -orbit. 5 Prove that U f and U ˆ Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 19 / 23

F q -linear sets of PG ( 1 , q 4 ) of rank 4 f are in the same Γ L ( 2 , q 4 ) -orbit iff there exist A , B , C , D ∈ F q 4 , AD − BC � = 0, 6 U f and U ˆ and σ = p k , Classes and equivalence of linear sets in PG ( 1 , q n ) Giuseppe Marino Irsee 2017 20 / 23