arXiv:2102.11790v1 [math.CO] 23 Feb 2021 iwda eeaiaino 3 hoe ] e lo[,Pro [2, also res see a 5], with Theorem start [3, We of idea. generalization this a explore as and viewed extend we paper this noainOc KI,gatn.K145.Tescn autho Innovatio second and The Development Research, 124950. National K the parti no. of the grant 120154 acknowledges NKFIH, author – third Develop Office the Research, Innovation and National first the The by 132463. and Technology and vation ihpolm nve f(i,frasre nac e 1 n f and [1] points) see arcs 2 on mod survey 0 a in for t it (ii), to of meets added view line be in every can problems point with (i.e. intersection way their this (ii) “regular” or 1; degree of n ieprec aallcas(hc s1scn) o,it Now, interpret be 1-secant). can is which (which m concurrent, class it are parallel lines class: each (exceptional) line per parallel every line in one 2 mod numbers intersection h usae fteabetsae nti ae erestrict we paper AG(2 this plane In affine space. ambient the of subspaces the regular” t structures. very and symmetric) “intersection-wise geometry; an ambient that the stating of results, subspaces “reg the very typically with str are These properties etc.) group. subfield, geometries symmetry a finite over large subgeometries of a history admitting the structures in i.e. motivations key the of One Introduction 1 euaiytesle.Frisac,i AG(2 i in reason instance, “corr the For be that may themselves. guess intersections regularity may non-regular the one then i.e. structure, classes parallel many for lotallnso aallcasi h aenme fpoin of number same hap the well in may class It parallel a papers.) of separate lines in all come almost will spaces other to ∗ h rtato a upre ythe by supported was author first The aua etse st netgt on es hc behav which sets, point investigate to is step next natural A ihecpinlitreto ubr r called are numbers intersection exceptional with nesc u e ntesm ubro ons(osbymod (possibly points plane of affine number a same almost desarguesian that the a such in lines, of set of sets our classes intersect point parallel many) investigate regula (sometimes: we “almost some have paper they this instance for In sense, (well-defined) some in eaiu fteerntn lines. renitent these of behaviour hr r ayeape o on esi nt emty hc beh which geometry, finite in sets point for examples many are There q , ,where ), ec sj´k ee zka,Zus Weiner Zsuzsa Sziklai, Csajb´ok, Peter Bence q sapwro h prime the of power a is NP1- e ainlEclec rga fteMnsr f Ministry the of Program Excellence National New UNKP-19-4 ´ eietlines Renitent Abstract q , ), 1 renitent q udo Hungary. of Fund n lspoto h ainlRsac,Dvlpetand Development Research, National the of support al vn naco size of arc an even, etadInvto ffie–NFH rn o PD no. grant NKFIH, – Office Innovation and ment p e utb n ntels fte(classical, the of list the on one be must set Tentrlbtntovosextensions obvious not but natural (The . ce” ra es hyas oss some possess also they least at or ected”, lr hnyucnie hi intersection their consider you when ular” n epoerslso h (regular) the on results prove we and , sgaeu o h ata upr fpoetK project of support partial the for grateful is r da i hyare they (i) as ed usle opitst fadesarguesian a of sets point to ourselves cue qarc,Hriinvarieties, Hermitian (quadrics, uctures oiin2 n 9 eak7]. Remark [9, and 2] position esec iei r2pit,except points, 2 or 0 in line each eets eoiia on e,wihbecomes which set, point original he stesuyo ymti structures, symmetric of study the is eeeitmn “classification-type” many exist here ses opoeta hs 1-secant these that prove to easy is “ e p s(osbymod (possibly ts or l,wihi rm[]adi a be can it and [5] from is which ult, htorpitsthsahidden a has set point our that s hr r ayppr dealing papers many are There . h hrceitc.Telines The characteristic). the , e,ta u on e intersects set point our that pen, almost k ”ln-nescinnumbers. line-intersection r” llnso n aallclass parallel one of lines ll (mod o hc hr exist there which for , v ams regularly” “almost ave eual”wt epc to respect with regularly” ∗ q p a lotregular almost has 1 + utst e 1] In [10]. see multisets ) onso ulcurve dual a of points p .I thappens it If ). rInno- or Lemma 1.1 (Lemma of renitent lines [5]). Let T be a point set of AG(2,q). A line ℓ with slope d is called renitent if there exists an integer md such that |ℓ ∩ T | 6≡ md (mod p) but every other line with slope d meet T in md modulo p points. The renitent lines are concurrent. Now we define renitent lines in the following, more general setting and then prove various generalizations of the lemma above.

Definition 1.2. Let M be a multiset of AG(2,q). For some integer λ ≤ (q − 1)/2 a direction (d) is called (q − λ)-uniform if there are at least (q − λ) affine lines with slope d meeting M in the same number of points modulo p. This number will be called the typical intersection number at (d). The rest of the lines with direction (d) will be called renitent. A sharply (q − λ)-uniform direction (d) is a direction incident with exactly (q − λ) affine lines meeting M in the same number of points modulo p.

In the definition above different directions might have different typical intersection numbers. Note that these numbers are uniquely determined for each (q − λ)-uniform direction because of the assumption λ ≤ (q − 1)/2. Under some conditions, we are able to prove that the renitent lines are contained in an algebraic envelope of relatively small class (a curve of the dual plane of relatively small degree), as it is in Theorem 2.1 below.

Theorem 2.1 Take a multiset T of AG(2,q) and let Eλ denote a set of at most q directions which are (q − λ)-uniform such that the following hold:

(i) 0 < λ ≤ min{q − 2,p − 1},

(ii) for each (d) ∈Eλ the renitent lines meet T in the same number, say td, of points modulo p,

(iii) for each (d) ∈Eλ if md denote the typical intersection number at direction (d), then td − md modulo p does not depend on the choice of (d).

Then the renitent lines with direction in Eλ are contained in an algebraic envelope of class λ.

In Theorem 2.2, we remove the condition on the size of Eλ and give a more general formulation of Theorem 2.1. The proof relies on k-th power sums and the Newton–Girard formulas. If we are more permissive with the renitent lines of T then the class of the algebraic envelope might increase. To prove the following result we apply a “weighted” version of the Newton–Girard formulas, see Lemma 3.1.

Theorem 3.3 Take any T ⊆ AG(2,q) and an integer 0 < λ ≤ (q − 1)/2. Let Eλ denote a set of (q − λ)-uniform directions of size at most q. The renitent lines with slope in Eλ are contained in an algebraic envelope of class λ2. Furthermore, if a direction is (q − λ)-uniform, but not sharply uniform, then the line pencil centered at that direction is fully contained in the envelope. At Section 4, we show how to apply the resultant method, cf. [10], with the help of a polynomial which can detect renitent lines at each uniform direction.

2 Envelopes of small class

In this section our aim is to show that there is an envelope of relatively small class containing the renitent lines with slope in a given subset Fλ of (q − λ)-uniform directions. There exist at least λ common lines of such an envelope and a line pencil centered at a sharply (q − λ)-uniform direction.

2 Of course, the union of the line pencils centered at the (q − λ)-uniform directions is an envelope containing these renitent lines. So the minimal class of such an envelope is at least min{|Fλ|, λ}. Any line set of size at most |Fλ|λ is contained in an algebraic envelope of class at most ⌈ 2|F |λ⌉− 1. Indeed, a 3-variable homogeneous polynomial h of degree d has d+2 coefficients. p λ 2
For any point P ∈ PG(2,q), the condition that h vanishes at P is equivalent to a linear equation for these coefficients. If d+2 is larger than the size of a point set S, then there is a non-trivial solution 2
for the homogeneous system of equations which corresponds to the condition that the points of S are zeroes of h. In Theorem 2.1, we show that no matter of the size of Fλ, with some conditions on the renitent lines we can always construct an envelope of class λ containing all renitent lines with slope in Fλ. The results of this section will rely on the Newton–Girard formulas. From now on, we coordinatize the projective plane PG(2,q) with homogeneous coordinates over GF(q) in such a way that the line at infinity is [0 : 0 : 1], while the X- and the Y -axes are [0 : 1 : 0] and [1 : 0 : 0], resp. The direction (d), where d ∈ Fq, means the point (1 : d : 0) on the line at infinity. The slope of the affine lines incident with (d) is d. Later in this section, we will be able to remove the condition |Eλ| ≤ q from Theorem 2.1, see Remark 2.3.

Theorem 2.1. Take a multiset T of AG(2,q) and let Eλ denote a non-empty set of at most q directions which are (q − λ)-uniform such that the following hold: (i) 0 < λ ≤ min{q − 2,p − 1},

(ii) for each (d) ∈Eλ the renitent lines meet T in the same number, say td, of points modulo p,

(iii) for each (d) ∈Eλ if md denote the typical intersection number at direction (d), then td − md modulo p does not depend on the choice of (d).

Then the renitent lines with direction in Eλ are contained in an algebraic envelope of class λ.

Proof. First we show that the number of renitent lines is the same at each direction (d) of Eλ. Let (d) and (e) denote two directions in Eλ which are sharply (q−λd)-uniform and sharply (q−λe)-uniform, respectively. Then

(q − λd)md + λdtd ≡|T|≡ (q − λe)me + λete (mod p), hence λd(td − md) ≡ λe(te − me) (mod p).

By assumption (iii), td − md ≡ te − me (mod p) and td 6≡ md (mod p), thus λd ≡ λe (mod p). Then λd = λe follows from the fact that 0 ≤ λe, λd ≤ λ ≤ p − 1. From now on, we may assume that the directions in Eλ are sharply (q − λ)-uniform. Since |Eλ| ≤ q, we may assume (0 : 1 : 0) ∈/ Eλ. For each (1 : d : 0) ∈ Eλ put (0 : α1(d) : 1), (0 : α2(d) : 1),..., (0 : αλ(d) : 1) for the points of the Y -axis on the renitent lines with slope d. s Put s := |T | and T = {(ai : bi : 1)}i=1. Next define the polynomials s π (V ) := (b − a V )k ∈ F [V ] k X i i q i=1 of degree at most k. The line joining (1 : d : 0) and (ai : bi : 1) meets the Y -axis at the point (0 : bi − aid : 1), hence for each (1 : d : 0) ∈Eλ the multiset

s Md := {(bi − aid)}i=1

3 contains md modulo p copies of Fq and c modulo p further copies of αi(d) for 1 ≤ i ≤ λ, where k c ∈ {1,...,p−1} is an integer such that c ≡ td −md (mod p). Since F γ = 0 for 0 ≤ k ≤ q −2 Pγ∈ q and since πk(d) is the k-th power sum of Md, for 0 ≤ k ≤ q − 2 it holds that for (1 : d : 0) ∈Eλ

λ π (d)= c α (d)k. (1) k X i i=1

Denote by σi(X1,...,Xλ) the i-th elementary symmetric polynomial in the variables X1,...,Xλ. Also, for any integer i ≥ 0 and d ∈Eλ put

σi(d)= σi(α1(d),...,αλ(d)).

For p − 1 ≥ j ≥ 1 define the following polynomial of degree at most j:

j ni j (−πi(V )/c) F Mj(V ) := (−1) n ∈ q[V ]. X Y ni!i i n1+2n2+...+jnj =j i=1 n1,n2,...,nj >0

Then for min{q − 2,p − 1}≥ j ≥ 1 from (1) and from the Newton-Girard identities it follows that Mj(d)= σj(d) for each (1 : d : 0) ∈Eλ. Consider the affine curve of degree λ defined by

λ λ−1 λ−2 λ−1 λ f(U, V ) := U − M1(V )U + M2(V )U − . . . + (−1) Mλ−1(V )U + (−1) Mλ(V ).

Then the projective curve of degree λ defined by the equation g(U, V, W ) := W λf(U/W, V/W ) contains the point (αi(d) : d : 1) for each d ∈Eλ and 1 ≤ i ≤ λ. Indeed,

λ λ−1 λ−2 λ−1 λ g(U, d, 1) = U − σ1(d)U + σ2(d)U − . . . + (−1) σλ−1(d)U + (−1) σλ(d)=

λ (U − α (d)). Y i i=1

It follows that the lines [d : −1 : αi(d)] are contained in an algebraic envelope of class λ.

Theorem 2.2. Take a multiset T of AG(2,q) and let Fλ denote a set of (q −λ)-uniform directions. For each (d) ∈ Fλ denote the typical intersection number by md and denote the intersection numbers F of the renitent lines by td,1,td,2,...,td,λd , for some 0 < λd ≤ λ. For c ∈ p \ {0} define the integers λd,i(c) ∈ {1,...,p − 1} such that

cλd,i(c) ≡ td,i − md (mod p) and assume that λd Λ (c) := λ (c) ≤ min{q − 2,p − 1} (2) d X d,i i=1 holds for each (d) ∈ Fλ (note that the sum is taken over natural numbers). Then Λ(c) := Λd(c) does not depend on d and the renitent lines with direction in Fλ are contained in an algebraic envelope of class Λ(c). Moreover, the intersection multiplicity of this envelope with the pencil centered at (d) at a renitent line incident with (d) and with intersection number td,i is λd,i(c).

4 Proof. Throughout the proof, we will fix c and so neglect it from λd,i(c), Λd(c) and Λ(c). As before, first we show that Λd does not depend on (d) of Fλ. Let (d) and (e) denote two directions in Fλ and define Λd and Λe as in (2). Then

qmd + Λdc ≡|T|≡ qme + Λec (mod p), hence Λd ≡ Λe (mod p).

Then Λd = Λe follows from the fact that 0 < Λe, Λd ≤ p − 1. First we prove the assertion for any Eλ ⊆ Fλ such that |Eλ| ≤ q. In this case, then we may assume(0 : 1 : 0) ∈E/ λ. For each (1 : d : 0) ∈Eλ put (0 : α1(d) : 1), (0 : α2(d) : 1),..., (0 : αλd (d) : 1) for the points of the Y -axis on the renitent lines with slope d. s Put s := |T | and T = {(ai : bi : 1)}i=1. Next define the polynomials

s π (V ) := (b − a V )k ∈ F [V ] k X i i q i=1 of degree at most k. For each (1 : d : 0) ∈Eλ the multiset

s Md := {(bi − aid)}i=1 contains md modulo p copies of Fq and td,i − md modulo p further copies of αi(d) for 1 ≤ i ≤ λd. k Since F γ = 0 for 0 ≤ k ≤ q −2 and since πk(d) is the k-th power sum of Md, for 0 ≤ k ≤ q −2 Pγ∈ q it holds that for (1 : d : 0) ∈Eλ λd λd,i π (d)= c α (d)k. (3) k X X i i=1 j=1

Denote by σi(X1,...,XΛ) the i-th elementary symmetric polynomial in the variables X1,...,XΛ. Also, for any integer i ≥ 0 and d ∈Eλ put

σi(d)= σi(α1(d),...,α1(d),...,αj(d),...,αj(d),...,αλd (d),...,αλd (d)).

λ 1 times λ times λ times | d, {z } | d,j{z } | d,λd{z } For p − 1 ≥ j ≥ 1 define the following polynomial of degree at most j:

j ni j (−πi(V )/c) F Mj(V ) := (−1) n ∈ q[V ]. X Y ni!i i n1+2n2+...+jnj =j i=1 n1,n2,...,nj >0

Then for min{q − 2,p − 1}≥ j ≥ 1 from (3) and from the Newton–Girard identities it follows that Mj(d)= σj(d) for each (1 : d : 0) ∈Eλ. Consider the affine curve of degree Λ defined by

Λ Λ−1 Λ−2 Λ−1 Λ f(U, V ) := U − M1(V )U + M2(V )U − . . . + (−1) MΛ−1(V )U + (−1) MΛ(V ).

Then the projective curve of degree Λ defined by the equation g(U, V, W ) := W Λf(U/W, V/W ) contains the point (αi(d) : d : 1) for each d ∈Eλ and 1 ≤ i ≤ λd, with multiplicity λd,i. Indeed,

Λ Λ−1 Λ−2 Λ−1 Λ g(U, d, 1) = U − σ1(d)U + σ2(d)U − . . . + (−1) σΛ−1(d)U + (−1) σΛ(d)=

5 λd (U − α (d))λd,i . Y i i=1

It follows that the point (αi(d) : d : 1) lies on the curve defined by f. Note that (V − d) cannot divide f(U, V ) and hence the “horizontal” line [0 : 1 : −d] cannot be a component of the curve defined by f. The intersection multiplicity of f and [0 : 1 : −d] is λd,i at the point (αi(d) : d : 1). It follows that the lines [d : −1 : αi(d)] are contained in an algebraic envelope of class Λ and the intersection multiplicity of this envelope with the pencil centered at (1 : d : 0) is λd,i at the line [d : −1 : αi(d)]. Now assume that |Fλ| = q + 1. Apply the argument above for two distinct subsets of Fλ, both ′ of them of size q. Denote them by Eλ and Eλ and denote the corresponding dual curves of degree Λ by C and C′, respectively. Our aim is to prove that these two curves coincide. Put C = H·A and C′ = H·A′, where A and A′ do not have a common component. Denote the degree of H by h and suppose for the contrary that h< Λ. ′ For (d) ∈Eλ ∩Eλ denote the line [0 : 1 : −d] by ℓd and the point (αi(d) : d : 1) by Pd,i. Recall ′ that ℓd cannot be a component of C or C . The intersection multiplicity I(C∩ ℓd, Pd,i)= λd,i equals I(H∩ ℓd, Pd,i)+I(A∩ ℓd, Pd,i), thus

I(A∩ ℓd, Pd,i)= λdi − I(H∩ ℓd, Pd,i), and by similar reasons ′ I(A ∩ ℓd, Pd,i)= λdi − I(H∩ ℓd, Pd,i). ′ By [7, Lemma 10.4] it follows that I(A∩A , Pd,i) ≥ λdi − I(H∩ ℓd, Pd,i). Then

λd λd λd λd I(A∩A′, P ) ≥ λ − I(H∩ ℓ , P ) = Λ − I(H∩ ℓ , P ) ≥ Λ − h, (4) X d,i X di X d d,i X d d,i i=1 i=1 i=1 i=1

λd ′ since I(H∩ ℓd, Pd,i) ≤ deg H = h. The inequality (4) holds for each (d) ∈Eλ ∩E and hence Pi=1 λ

λd ′ ′ I(A∩A , P ) ≥ I(A∩A , Pd,i) ≥ (q − 1)(Λ − h). X ′ X ′ X P ∈A∩A (d)∈Eλ∩Eλ i=1

′ ′ On the other hand, By B´ezout’s theorem P ∈A∩A′ I(A∩A , P ) ≤ deg A · deg A = (Λ − h)(Λ − h). This contradiction proves h = Λ, thatP is, C = C′.

Remark 2.3. If in Theorem 2.2 we assume td,1 = td,2 = . . . = td,λd =: td for each (d) ∈ Fλ, we also assume that td−md does not depend on the choice of d, and further assume 0 < λ ≤ min{q−2,p−1}, then with the choice c ≡ td − md (mod p) we obtain Theorem 2.1 without the restriction |Eλ|≤ q. Remark 2.4. If |T| ≡ 0 (mod p) then Theorem 2.2 cannot be applied. Indeed, in that case for each (d) ∈ Fλ, λd (t − m ) ≡ 0 (mod p) X d,i d i=1

λd and hence Λd = i=1 λd,i ≡ 0 (mod p), which is not possible if Λd ≤ p − 1 (by definition λd,i > 0 for each i). P

6 Remark 2.5. For c = 1, 2,...,p − 1, the values of Λd(c) give different residues modulo p and this residue is the same for every choice of d. However, it is possible that for some of the directions, Λd(c) ≤ min{q − 2,p − 2} does not hold. Indeed, put for example p = 5, λ = 2 and assume that the typical intersection number is 0 at each direction. Also, assume that both of the two renitent lines with direction (d1) ∈ Fλ meet T in 1 modulo p points and the two renitent lines with direction (d2) ∈ Fλ meet T in 3 and in 4 points modulo p. Then, with c = 1 we obtain Λd1 (c) = 2 and Λd2 (c) = 7, so the result cannot be applied. On the other hand, with c = 3 we obtain Λd1 (c) = Λd2 (c) = 4 and hence the result can be applied if q > 5. It shows that sometimes it might be convenient to choose Fλ not as the set of all (q − λ)-renitent lines, but as a subset of them.

Example 2.6. In Theorem 2.2 put λ = 3 and assume that md = 1 and the renitent lines meet T modulo p in the multiset {3, 3, 5} for each (d) ∈ Fλ. Note that this implies p 6= 2. With the choice c = 2 it follows that the renitent lines are contained in a curve of degree Λ = (3 − 1)/2+ (3 − 1)/2+ (5 − 1)/2 = 4 whenever 4 ≤ min{q − 2,p − 1}. One might think that to obtain a curve of the lowest degree, the best option is to chose c as the greatest common divisor of the values td,i − md. The next example refute this belief.

Example 2.7. In Theorem 2.2, put λ = 2, p = 13, and assume that md = 1 and the renitent lines meet T modulo 13 in the multiset {2, 8} for each (d) ∈ Fλ. With the choice c = 1 it follows that the renitent lines are contained in a curve of degree Λ = (2 − 1) + (8 − 1) = 8. With the choice c = 7 it follows that the renitent lines are contained in a curve of degree Λ = 1/7 + 7/7=2+1=3.

3 The general case

First we prove a recursion connecting elementary symmetric polynomials with “weighted” power sums.

Lemma 3.1. Let σk = σk(X1,...,Xλ) denote the k-th elementary symmetric polynomial in the variables X1,...,Xλ. For some field elements c1, c2,...,cλ put λ P = P (X ,...,X )= c Xk. k k 1 λ X i i i=1 Then for any integer j ≥ 0 it holds that λ+1 Pλ+j = Pλ+j−1σ1 − Pλ+j−2σ2 + . . . + (−1) Pjσλ. Proof. Note that

λ Y j (Y − X )= Y λ+j − Y λ+j−1σ + Y λ+j−2σ − . . . + (−1)λY jσ . Y i 1 2 λ i=1 It follows that λ+j λ+j−1 λ+j−2 λ j c1(X1 − X1 σ1 + X1 σ2 − . . . + (−1) X1 σλ) = 0, λ+j λ+j−1 λ+j−2 λ j c2(X2 − X2 σ1 + X2 σ2 − . . . + (−1) X2 σλ) = 0, . . λ+j λ+j−1 λ+j−2 λ j cλ(Xλ − Xλ σ1 + Xλ σ2 − . . . + (−1) Xλσλ) = 0. Summing up both sides above yields the assertion.

7 Lemma 3.2. As before, put

λ P = P (X ,...,X )= c Xk, k k 1 λ X i i i=1 and define the λ × λ matrix

P P ... P0  λ−1 λ−2  Pλ Pλ−1 ... P1 H = H(X1,...,Xλ)=  . . . .  .  . . . .    P2λ−2 P2λ−3 ... Pλ−1

λ(λ−1)/2 2 Then det H = (−1) c1c2 ...cλ (Xi − Xj) . Q1≤i

λ−1 λ−2 c1 c2 ... cλ X1 X1 . . . 1    λ−1 λ−2  c1X1 c2X2 ... cλXλ X2 X2 . . . 1 H =  .   .  .  .   .   λ−1 λ−1 λ−1  λ−1 λ−2  c1X1 c2X2 ... cλXλ  Xλ Xλ . . . 1

Theorem 3.3. Take any T ⊆ AG(2,q) and an integer 0 < λ ≤ (q − 1)/2. Let Eλ denote a set of (q − λ)-uniform directions of size at most q. The renitent lines with slope in Eλ are contained in 2 an algebraic envelope of class λ . Furthermore, if a direction of Eλ is not sharply (q − λ)-uniform, then the line pencil centered at that direction is fully contained in the envelope.

Proof. For each (1 : d : 0) ∈Eλ denote by λd the number of renitent lines with slope d and denote by (0 : α1(d) : 1), (0 : α2(d) : 1),..., (0 : αλd (d) : 1) the points of the Y -axis on these lines. Also, s denote the typical intersection number at (d) by md. Put s := |T | and T = {(ai : bi : 1)}i=1. Next define the polynomials s π (V ) := (b − a V )k ∈ F [V ] k X i i q i=1 of degree at most k. For any (1 : d : 0) ∈Eλ the multiset

s Md := {(bi − aid)}i=1 contains md modulo p copies of Fq and ci(d) 6≡ 0 modulo p further copies of αi(d) for 1 ≤ i ≤ λd. k Since g∈Fq g = 0 for 0 ≤ k ≤ q −2 and since πk(d) is the k-th power sum of Md, for 0 ≤ k ≤ q −2 it holdsP that λd π (d)= c (d)α (d)k. (5) k X i i i=1

Note that πk(d) is as Pk(λ1, λ2, . . . , λλd ) in Lemma 3.1. For any integer i ≥ 0 and (1 : d : 0) ∈ Eλ put

σi(d)= σi(α1(d),...,αλd (d)).

For j ≥ 0, q − 2 ≥ λd + j and (1 : d : 0) ∈Eλ Lemma 3.1 yields

λ πλd+j(d)= πλd+j−1(d)σ1(d) − πλd+j−2(d)σ2(d)+ . . . − (−1) πj(d)σλd (d). (6)

8 Define π (V ) π (V ) ... π0(V )  λ−1 λ−2  πλ(V ) πλ−1(V ) ... π1(V ) H(V )=  . . . .  .  . . . .    π2λ−2(V ) π2λ−3(V ) ... πλ−1(V )

Then for each (1 : d : 0) ∈Eλ, since 2λ − 1 ≤ q − 2, by (6) applied for λ − λd ≤ j ≤ 2λ − λd − 1, we obtain σ1(d) π (d)    λ  −σ2(d) πλ+1(d) H(d)  .  =  .  . (7)  .   .   λ+1    (−1) σλ(d) π2λ−1(d)

Denote by Hi(V ) the matrix obtained from H(V ) by replacing its i-th column with

T C(V ) := (πλ(V ),πλ+1(V ),...,π2λ−1(V )) .

By Lemma 3.2, if (1 : d : 0) ∈ Eλ is sharply (q − λ)-uniform, then the determinant of H(d) is λ(λ−1)/2 λ 2 (−1) i=1 ci(d) 1≤i

f(U, V ) :=

λ λ−1 λ−2 λ−1 λ M(V )U − M1(V )U + M2(V )U − . . . + (−1) Mλ−1(V )U + (−1) Mλ(V ).

When λd = λ, then (αi(d), d) is a point of C for each i and (1 : d : 0) ∈ Eλ. Indeed, in this case f(U, d) equals

λ λ−1 λ−2 λ−1 λ M(d)(U − σ1(d)U + σ2(d)U − . . . + (−1) σλ−1(d)U + (−1) σλ(d)) =

λ M(d) (U − α (d)). Y i i=1

Now consider (1 : d : 0) ∈Eλ such that λd < λ. To show that the pencil with carrier (d) is contained in the dual curve it is enough to prove that M(d)= M1(d)= . . . = Mλ(d) = 0. By (6) applied for 0 ≤ j ≤ λ − 1, in H(d) the column π (d)  λd  πλd+1(d)  .   .    πλd+λ−1(d) is the linear combination of the subsequent λd columns and hence det H(d)=0 and so M(d) = 0. By (7), the vector C(d) is the linear combination of the columns of H(d) and hence when H(d) is singular, then so does Hi(d) for each i. It follows that det Hi(d)= Mi(d) = 0 as we claimed.

From the last paragraphs of the previous proof the following is clear.

9 Corollary 3.4. Suppose that the assumptions of the previous theorem holds and also that there exists a sharply (q − λ)-uniform direction. Then there are at most λ2 − λ directions incident with less than λ renitent lines. More precisely, if λd denotes the number of renitent lines with slope d for every (d) ∈Eλ, then (λ − λ ) ≤ λ2 − λ. X d (d)∈Eλ Proof. Suppose for the contrary that there are at least λ2 − λ + 1 directions incident with less than λ renitent lines. Then the algebraic envelope of renitent lines contains the pencils of carriers with these directions. It follows that the renitent lines incident with sharply (q − λ)-uniform directions are contained in a dual curve of degree less than λ. We know that this dual curve does not contain the corresponding pencils (in these cases M(d) 6= 0 in the previous proof) and hence it cannot meet a pencil in λ lines, a contradiction. To see the more precise bound, assume that (d) ∈ Eλ is sharply (q − λd)-uniform for some λd < λ. As at the end of the previous proof, it can be seen that each of the first λ − λd columns of H(d) are linear combinations of the subsequent λd columns and hence the rank of H(d) is at most λd. Then for s > λd the s × s minors of H(d) have zero determinant. It follows that d is at least a 2 (λ − λd)-fold root of det H(V ), see [11, Lemma 2.3], which has degree at most λ − λ and the proof is completed.

4 The resultant method

The next result was developed in a series of papers by Sz˝onyi and Weiner [8, 11], see also [7] and the Appendix of [6] for the version that we cite here. Result 4.1 (Sz˝onyi–Weiner Lemma). Let f, g ∈ F[X,Y ] be polynomials over the arbitrary field F. deg f Assume that the coefficient of X in f is not 0 and for y ∈ F put ky = deg gcd(f(X,y), g(X,y)). Then for any y0 ∈ F (k − k )+ ≤ (deg f − k )(deg g − k ). 1 (8) X y y0 y0 y0 y∈F The main ingredient of the next proofs is how we define the polynomial g(X,Y ) in the above lemma. In order to be able to detect the renitent lines at the (q − λ)-uniform directions, in the definition of g we introduce an auxiliary polynomial h that we obtain by interpolation.

Theorem 4.2. Take a multiset M of AG(2,q), and fix an integer λ > 0. Let Eλ denote a set of (q − λ)-uniform directions, containing at least one sharply (q − λ)-uniform direction. If |Eλ| ≤ q 2 then there are at least λ(|Eλ| + 1 − λ) renitent lines and hence, at most λ − λ directions incident with less than λ renitent lines (counted with multiplicities, as in Corollary 3.4).

k Proof. Put |Eλ| = k, Eλ = {(1 : di : 0)}i=1 and M = {(ai : bi : 1)}i=1. Denote by mi the typical intersection number corresponding to the point (1 : di : 0). We will need the polynomial k q−1 F h(Y ) := i=1 mi(1 − (Y − di) ) ∈ q[Y ]. Then h(di) = mi for each (d) ∈ Eλ. Next define the polynomialsP f(X,Y ) := Xq − X, g(X,Y ) := (X + a Y − b )q−1 − |M| + h(Y ). X i i i

1 + Here α = max{0, α}. Note that g can be the zero polynomial as well, in that case deg f = ky = ky0 and the lemma claims the trivial 0 ≤ 0.

10 For a direction (y) ∈Eλ let ind(y) denote the number of renitent lines incident with (y). Then for any (y) ∈Eλ it holds that

ky := deg gcd(f(X,y), g(X,y)) = q − ind(y).

Then for any fixed R ∈Eλ, (8) gives

(ind R − ind(y)) ≤ (ind R − ind(y))+ ≤ ind R(ind R − 1). X X F (y)∈Eλ y∈ q

If ind R = λ, then, as in Corollary 3.4

kλ − ind(y) ≤ λ(λ − 1), X (y)∈Eλ

λ(k + 1 − λ) ≤ ind(y) X (y)∈Eλ

Theorem 4.3. Take a multiset M of AG(2,q), q > 2, and fix an integer λ> 0. Let Fλ denote the 2 set of (q − λ)-uniform directions. If |Fλ| > λ + λ then for each point R of the plane it holds that R is incident with at most λ or with at least |Fλ| + 1 − λ renitent lines.

Proof. First we prove the following. If Eλ is a set of at most q directions which are (q − λ)-uniform 2 and |Eλ| > λ + λ, then for each point R of the plane it holds that R is incident with at most λ or with at least |Eλ| + 1 − λ renitent lines with slope in Eλ. Consider AG(2,q) embedded in PG(2,q) as {(a : b : 1) : a, b ∈ GF(q)} and apply a collineation of PG(2,q) which maps the line [0 : 0 : 1] (the line at infinity) to the line [1 : 0 : 0] (the Y -axis) such that (0 : 1 : 0) is not in the image of Eλ. It may be further assumed that this collineation maps R to some point (1 : y0 : 0). Denote the image of M by {(ai : bi : 1)}i ∪ {(1 : zj : 0)}j and the image of Eλ by {(0 : ck : 1)}k. Let mk be the typical intersection number corresponding to the point (0 : ck : 1). We will |Eλ| q−1 need the polynomial h(X) = mk(1 − (X − ck) ) ∈ Fq[X]. Then h(ck) = mk for each Pk=1 k ∈ {1,..., |Eλ|}. Next define the polynomials

|Eλ| f(X,Y ) := (X − c ), Y k k=1

g(X,Y ) := (X + a Y − b )q−1 + (Y − z )q−1 − |M| + h(X). X i i X j i j For a point P ∈ PG(2,q) let ind P denote the number of renitent lines incident with P . Then for any y ∈ Fq it holds that

ky := deg gcd(f(X,y), g(X,y)) = |Eλ|− ind(1 : y : 0).

Then (8) gives (ind R − ind(1 : y : 0))+ ≤ ind R(q − 1 − |E | + ind R). X λ y∈Fq

11 Note that ind(1 : y : 0) ≤ |E |λ X λ y∈Fq and hence q ind R − |Eλ|λ ≤ ind R(q − 1 − |Eλ| + ind R), 2 0 ≤ ind R − ind R(|Eλ| +1)+ |Eλ|λ.

For ind R = λ + 1, or ind R = |Eλ|− λ we have

2 2 ind R − ind R(|Eλ| +1)+ |Eλ|λ = λ + λ − |Eλ|,

2 which is less than 0 since |Eλ| > λ + λ. This proves ind R ≤ λ or ind R ≥ |Eλ|− λ + 1. Of course, if |Fλ|≤ q then one can take Eλ = Fλ and this proves the theorem. 2 Now assume q +1 = |Fλ| > λ + λ and take an affine point R. We have to show that R is incident with at most λ or with at least q +2−λ renitent lines. Define Eλ as any subset of directions 2 of size q. Note that q +1= |Fλ|= 6 λ + λ + 1 because this would imply q = λ(λ + 1) and hence 2 λ = 1 and q = 2, which we excluded. It follows that |Eλ| = q > λ + λ and hence the arguments above show that R is incident with at most λ + 1 renitent lines or with at least q + 1 − λ renitent lines. We have to exclude the cases when R is incident with exactly λ + 1 renitent lines or with exactly q + 1 − λ renitent lines. If the former case holds, then take a direction S such that RS is not renitent and define Eλ as ℓ∞ \ {S}. Then the renitent lines incident with R have slopes in Eλ, a contradiction since there are more than λ but less than q + 1 − λ of them. If the latter case holds, then take a direction S such that RS is renitent and, as before, define Eλ as ℓ∞ \ {S}. Then there are exactly q − λ renitent lines incident with R with slopes in Eλ, a contradiction since this number should be at least q + 1 − λ or at most λ (recall that q +1= |Fλ| > λ2 + λ).

With some further efforts, the earlier methods by Sz˝onyi and Weiner [8], [11], see also [7], [4], also provide an alternative, but more laborious, route to reach similar results as in Section 3.

References

[1] S. Ball, M. Lavrauw: Arcs in finite projective spaces, EMS Surveys in Mathematical Science 6 (2019), 133–172.

[2] A. Blokhuis: Characterization of seminuclear sets in a finite projective plane, J. Geom. 40 (1991), 15–19.

[3] A. Blokhuis, T. Szonyi:˝ Note on the structure of semiovals, Discrete Math. 106/107 (1992), 61–65.

[4] A. Blokhuis, T. Szonyi,˝ Zs. Weiner: On Sets without Tangents in Galois Planes of Even Order, Designs, Codes and Cryptography 29 (2003), 91–98.

[5] B. Csajbok,´ Zs. Weiner: Generalizing Korchm´aros–Mazzocca arcs, to appear in Com- binatorica, (2021), https://doi.org/10.1007/s00493-020-4419-z

[6] T. Heger:´ Some graph theoretic aspects of finite geometries, PhD Thesis, E¨otv¨os Lor´and University, 2013, http://heger.web.elte.hu//publ/HTdiss-e.pdf

12 [7] P. Sziklai: Polynomials in Finite Geometry, Manuscript available online at https://web.cs.elte.hu/∼sziklai/polynom/poly08feb.pdf

[8] T. Szonyi:˝ On the Embedding of (k,p)-Arcs in Maximal Arcs, Designs, Codes and Cryp- tography 18 (1999), 235–246.

[9] T. Szonyi:˝ On the number of directions determined by a set of points in an affine Galois plane, J. Combin. Theory Ser. A 74 (1996), 141–146.

[10] T. Szonyi,˝ Zs. Weiner: Stability of k mod p multisets and small weight codewords of the code generated by the lines of PG(2,q), J. Combin. Theory Ser. A 157 (2018), 321–333.

[11] Zs. Weiner: On (k,pe)-arcs in Desarguesian planes, Finite Fields Appl. 10 (2004), 390– 404.

Bence Csajb´ok ELKH–ELTE Geometric and Algebraic Combinatorics Research Group ELTE E¨otv¨os Lor´and University, Budapest, Hungary Department of Computer Science 1117 Budapest, P´azm´any P. stny. 1/C, Hungary [email protected] Peter Sziklai E¨otv¨os Lor´and University, Budapest, Hungary Department of Computer Science 1117 Budapest, P´azm´any P. stny. 1/C, Hungary, and R´enyi Institute of Mathematics 1053 Budapest, Re´altanoda u. 13-15. [email protected] Zsuzsa Weiner ELKH–ELTE Geometric and Algebraic Combinatorics Research Group, 1117 Budapest, P´azm´any P. stny. 1/C, Hungary [email protected] and Prezi.com H-1065 Budapest, Nagymez˝outca 54-56, Hungary

13