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A lower bound on the size of linear sets in PG(1, qn)

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, qn) is an Fq-linear set of rank k if it equals a set LU for some Fq-vector subspace U of Frn

q of dimension k, where

LU = {uqn | u ∈ U∗}.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 1/21

Linear sets

Definition

Let k ≥ 1 and r ≥ 2. A point set in PG(r − 1, qn) is an Fq-linear set of rank k if it equals a set LU for some Fq-vector subspace U of Frn

q of dimension k, where

LU = {uqn | u ∈ U∗}.

Definition

The weight of the point P in LU is defined as wt(P) = dimq(vqn ∩ U).

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 1/21

Geometrical point of view

PG(r − 1, qn) → PG(nr − 1, q) (field reduction)

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 2/21

Geometrical point of view

PG(r − 1, qn) → 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, qn) September 12, 2018 2/21

Geometrical point of view

PG(r − 1, qn) → 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, qn) September 12, 2018 2/21

Geometrical point of view

PG(r − 1, qn) → 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, qn) September 12, 2018 2/21

Linear sets and linearized polynomials

Lemma

Let LU be an Fq-linear set of rank k in PG(1, qn), k ≤ n, not containing the point (0, 1)qn, then LU = {(x, f(x))qn|x ∈ V∗} for some vector subspace V ⊂ Fqn of dimension k and some Fq-linear map f : V → Fqn.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 3/21

Linear sets and directions

Lemma

The number of points of L = {(x, f(x))qn|x ∈ V∗}, where V is a vector subspace of Fqn and f : V → Fqn is an Fq-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, qn) September 12, 2018 4/21

Minimum size of linear sets in PG(1, qn)?

Is there is lower bound on the size of a linear set in PG(1, qn)?

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 5/21

Direction determined by a function

Theorem (S. Ball, 2003)

Let f : Fq → Fq be a function. Let N be the number of directions determined by f. Let s = pe 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

2

≤ N ≤ q + 1; (ii) Fs is a subfield of Fq and q

s + 1 ≤ N ≤ q−1 s−1;

(iii) s = q and N = 1. Moreover, if s > 2, then the graph of f is Fs-linear.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 6/21

Directions determined by a function

Let U ⊂ Fqn be a k-dimensional Fq vector space. Let f : U → Fn

q

be an Fq linear function. Is there a lower bound on the number

• f directions determined by f?

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 7/21

Some nice polynomials ...

LU = {(x, f(x)qn | x ∈ V∗} A = {(1, x, f(x))qn | x ∈ V} R(X, Y) =

• x∈V

(X − xY + f(x)) deg R(X, Y) = qk . R(X, Y) = Xqk +

qk

• j=1

σj(Y)Xqk−j . deg σj(Y) ≤ j .

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 8/21

Usual arguments

For an affine point set of size qn: If y ∈ Fqn is a slope that is not determined, then R(X, y) = Xqn − X . strong information on σj(Y)′s.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 9/21

Usual arguments

in our case If y is a slope not coming from a point in LU, then R(X, y) | Xqn − X . strong information on σj(Y)′s.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 10/21

The shape of R(X, Y)

Lemma

Let P = (x0, f(x0))qn be a point of weight j in LU = {(x, f(x)qn | x ∈ V∗}, then R(X, y0) with y0 = f(x0)/x0 is of the form R(X, y0) =

qk−j

• i=1

(X − αi)qj, for distinct αi ∈ Fqn.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 11/21

The shape of R(X, Y)

Lemma

If A = {(1, x, f(x))qn | x ∈ V}, where V is an Fq-vector subspace of Fqn of dimension k and f : V → Fqn is an Fq-linear map, then the Rédei polynomial of A is of the following shape: R(X, Y) = Xqk + σqk−qk−1(Y)Xqk−1 + σqk−qk−2(Y)Xqk−2 + . . . +σqk−1(Y)X .

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 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”.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 13/21

Euclidean division with remainder in Fqn[Y][X]: Xqn − X = R(X, Y)Q(X, Y) + r(X, Y) . so degX r(X, Y) < degX R(X, Y) H(X, Y) := −r(X, Y) − X . Since R(X, Y) is monic of degree qk, we can write Q(X, Y) = Xqn−qk +

qn−qk

• i=1

σ∗

i (Y)Xqn−qk−i .

(1)

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 14/21

A lemma on the degrees

Lemma

We have deg Q(X, Y) ≤ qn and deg r(X, Y) ≤ qn (where deg Q(X, Y) means the total degree). Furthermore, deg σ∗

i (Y) ≤ i.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 15/21

A lemma on the degrees

Lemma

We have deg Q(X, Y) ≤ qn and deg r(X, Y) ≤ qn (where deg Q(X, Y) means the total degree). Furthermore, deg σ∗

i (Y) ≤ i.

Corollary

We have degX H(X, Y) ≤ qk − 1. Let H(X, Y) =

qn

• i=0

hi(Y)Xqn−i , then deg hi(Y) ≤ i.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 15/21

A lower bound on the size of LU

Lemma

The the number of points in LU = {(x, f(x)qn | x ∈ V∗} is at least degX H(X, Y).

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 16/21

Final steps

Lemma

The number of points in an Fq-linear set is congruent to 1 mod q.

Theorem (DB and Van de Voorde)

Let LU = {(x, f(x))qn | x ∈ V∗}, where V has dimension k, be an Fq-linear set in PG(1, qn) of rank k which contains at least one point of weight one, then the size of LU is at least qk−1 + 1.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 17/21

The bound is sharp

Lemma

Let 2 ≤ k ≤ n. There exists an Fq-linear set of rank k in PG(1, qn) with qk−1 + 1 elements.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 18/21

Linear sets in PG(2, qn)

Theorem (DB and Van de Voorde)

Let L be an Fq-linear set of rank k in PG(2, qn) such that there is at least one line of PG(2, qn) meeting L in exactly q + 1 points, then L contains at least qk−1 + qk−2 + 1 points.

Lemma

Let 3 ≤ k ≤ n. There exists an Fq-linear set of rank k in PG(2, qn) with qk−1 + qk−2 + 1 elements.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 19/21

References

• S. Ball.

The number of directions determined by a function over a finite field.

• J. Combin. Theory Ser. A, 104(2):341–350, 2003.
• A. Blokhuis, S. Ball, A. E. Brouwer, L. Storme, and T

. Szőnyi. On the number of slopes of the graph of a function defined on a finite field.

• J. Combin. Theory Ser. A, 86(1):187–196, 1999.
• G. Bonoli and O. Polverino.

Fq-linear blocking sets in PG(2, q4).

• Innov. Incidence Geom., 2:35–56, 2005.
• Sz. L. Fancsali, P

. Sziklai, and M. Takáts. The number of directions determined by less than q points.

• J. Algebraic Combin., 37(1):27–37, 2013.
• M. Lavrauw and G. Van de Voorde.

On linear sets on a projective line.

• Des. Codes Cryptogr., 56(2-3):89–104, 2010.
• R. Lidl and H. Niederreiter.

Finite fields, volume 20 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P . M. Cohn. Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 20/21

References

• O. Ore.

On a special class of polynomials.

• Trans. Amer. Math. Soc., 35(3):559–584, 1933.
• O. Polverino.

Linear sets in finite projective spaces. Discrete Math., 310(22):3096–3107, 2010. P . Sziklai. On small blocking sets and their linearity.

• J. Combin. Theory Ser. A, 115(7):1167–1182, 2008.

Jan De Beule A lower bound on the size of linear sets in PG(1, qn) September 12, 2018 21/21