A lower bound on the size of linear sets in PG ( 1 , q n ) joint - PowerPoint PPT Presentation

A lower bound on the size of linear sets in PG ( 1 , q n ) joint work with Geertrui Van de Voorde Jan De Beule September 12, 2018

Linear sets Definition Let k ≥ 1 and r ≥ 2. A point set in PG ( r − 1 , q n ) is an F q -linear set of rank k if it equals a set L U for some F q -vector subspace U of F rn q of dimension k , where L U = {� u � q n | u ∈ U ∗ } . Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 1/21

Linear sets Definition Let k ≥ 1 and r ≥ 2. A point set in PG ( r − 1 , q n ) is an F q -linear set of rank k if it equals a set L U for some F q -vector subspace U of F rn q of dimension k , where L U = {� u � q n | u ∈ U ∗ } . Definition The weight of the point P in L U is defined as wt ( P ) = dim q ( � v � q n ∩ U ) . Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 1/21

Geometrical point of view PG ( r − 1 , q n ) → PG ( nr − 1 , q ) (field reduction) Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 2/21

Geometrical point of view PG ( r − 1 , q n ) → PG ( nr − 1 , q ) (field reduction) point → ( n − 1 ) -dimensional subspace Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 2/21

Geometrical point of view PG ( r − 1 , q n ) → PG ( nr − 1 , q ) (field reduction) point → ( n − 1 ) -dimensional subspace all points → Desarguesian spread S Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 2/21

Geometrical point of view PG ( r − 1 , q n ) → PG ( nr − 1 , q ) (field reduction) point → ( n − 1 ) -dimensional subspace all points → Desarguesian spread S linear set of rank k ← elements of S hit by a k − 1-dimensional subspace. Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 2/21

Linear sets and linearized polynomials Lemma Let L U be an F q -linear set of rank k in PG ( 1 , q n ) , k ≤ n, not containing the point � ( 0 , 1 ) � q n , then L U = {� ( x , f ( x )) � q n | x ∈ V ∗ } for some vector subspace V ⊂ F q n of dimension k and some F q -linear map f : V → F q n . Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 3/21

Linear sets and directions Lemma The number of points of L = {� ( x , f ( x )) � q n | x ∈ V ∗ } , where V is a vector subspace of F q n and f : V → F q n is an F q -linear map, is equal to the number of directions determined by the affine pointset A = {� ( 1 , x , f ( x )) � | x ∈ V } . Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 4/21

Minimum size of linear sets in PG ( 1 , q n ) ? Is there is lower bound on the size of a linear set in PG ( 1 , q n ) ? Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 5/21

Direction determined by a function Theorem (S. Ball, 2003) Let f : F q → F q be a function. 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: (i) s = 1 and q + 3 ≤ N ≤ q + 1 ; 2 s + 1 ≤ N ≤ q − 1 (ii) F s is a subfield of F q and q s − 1 ; (iii) s = q and N = 1 . Moreover, if s > 2 , then the graph of f is F s -linear. Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 6/21

Directions determined by a function Let U ⊂ F q n be a k -dimensional F q vector space. Let f : U → F n q be an F q linear function. Is there a lower bound on the number of directions determined by f ? Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 7/21

Some nice polynomials ... L U = {� ( x , f ( x ) � q n | x ∈ V ∗ } A = {� ( 1 , x , f ( x )) � q n | x ∈ V } � R ( X , Y ) = ( X − xY + f ( x )) x ∈ V deg R ( X , Y ) = q k . q k R ( X , Y ) = X q k + σ j ( Y ) X q k − j . � j = 1 deg σ j ( Y ) ≤ j . Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 8/21

Usual arguments For an affine point set of size q n : If y ∈ F q n is a slope that is not determined, then R ( X , y ) = X q n − X . strong information on σ j ( Y ) ′ s . Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) September 12, 2018 9/21

Usual arguments in our case If y is a slope not coming from a point in L U , then R ( X , y ) | X q n − X . strong information on σ j ( Y ) ′ s . September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 10/21

The shape of R ( X , Y ) Lemma Let P = � ( x 0 , f ( x 0 )) � q n be a point of weight j in L U = {� ( x , f ( x ) � q n | x ∈ V ∗ } , then R ( X , y 0 ) with y 0 = f ( x 0 ) / x 0 is of the form q k − j � ( X − α i ) q j , R ( X , y 0 ) = i = 1 for distinct α i ∈ F q n . September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 11/21

The shape of R ( X , Y ) Lemma If A = {� ( 1 , x , f ( x )) � q n | x ∈ V } , where V is an F q -vector subspace of F q n of dimension k and f : V → F q n is an F q -linear map, then the Rédei polynomial of A is of the following shape: R ( X , Y ) = X q k + σ q k − q k − 1 ( Y ) X q k − 1 + σ q k − q k − 2 ( Y ) X q k − 2 + . . . + σ q k − 1 ( Y ) X . September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 12/21

Alternative approach (Alternative compared with the original proof of Simeon) Inpired by a result of Fancsali, Sziklai, and Takáts on “The number of directions determined by less than q points”. September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 13/21

Euclidean division with remainder in F q n [ Y ][ X ] : X q n − X = R ( X , Y ) Q ( X , Y ) + r ( X , Y ) . so deg X r ( X , Y ) < deg X R ( X , Y ) H ( X , Y ) := − r ( X , Y ) − X . Since R ( X , Y ) is monic of degree q k , we can write q n − q k Q ( X , Y ) = X q n − q k + i ( Y ) X q n − q k − i . � σ ∗ (1) i = 1 September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 14/21

A lemma on the degrees Lemma We have deg Q ( X , Y ) ≤ q n and deg r ( X , Y ) ≤ q n (where deg Q ( X , Y ) means the total degree). Furthermore, deg σ ∗ i ( Y ) ≤ i. September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 15/21

A lemma on the degrees Lemma We have deg Q ( X , Y ) ≤ q n and deg r ( X , Y ) ≤ q n (where deg Q ( X , Y ) means the total degree). Furthermore, deg σ ∗ i ( Y ) ≤ i. Corollary We have deg X H ( X , Y ) ≤ q k − 1 . Let q n h i ( Y ) X q n − i , � H ( X , Y ) = i = 0 then deg h i ( Y ) ≤ i. September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 15/21

A lower bound on the size of L U Lemma The the number of points in L U = {� ( x , f ( x ) � q n | x ∈ V ∗ } is at least deg X H ( X , Y ) . September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 16/21

Final steps Lemma The number of points in an F q -linear set is congruent to 1 mod q. Theorem (DB and Van de Voorde) Let L U = {� ( x , f ( x )) � q n | x ∈ V ∗ } , where V has dimension k, be an F q -linear set in PG ( 1 , q n ) of rank k which contains at least one point of weight one, then the size of L U is at least q k − 1 + 1 . September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 17/21

The bound is sharp Lemma Let 2 ≤ k ≤ n. There exists an F q -linear set of rank k in PG ( 1 , q n ) with q k − 1 + 1 elements. September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 18/21

Linear sets in PG ( 2 , q n ) Theorem (DB and Van de Voorde) Let L be an F q -linear set of rank k in PG ( 2 , q n ) such that there is at least one line of PG ( 2 , q n ) meeting L in exactly q + 1 points, then L contains at least q k − 1 + q k − 2 + 1 points. Lemma Let 3 ≤ k ≤ n. There exists an F q -linear set of rank k in PG ( 2 , q n ) with q k − 1 + q k − 2 + 1 elements. September 12, Jan De Beule A lower bound on the size of linear sets in PG ( 1 , q n ) 2018 19/21