Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions
We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect Nonlinear) over…
Authors: Fern, o Hern, o
Pro of of a Conjecture on the Sequence of Exception a l Num b ers, Classifying Cycli c Co des and APN F unction s F ernando Hernando 1 Dep artment of Mathema tics University Col le ge Cork Ir eland Gary McGuire 2 Scho ol of Mathematic al Scienc es University Col le ge Dublin Ir eland Abstract W e prov e a conjecture that classifies exceptional num bers . Th is conjecture arises in t w o different w a ys, from cryptograph y and from co din g theory . An od d in tege r t ≥ 3 is said to b e exceptional if f ( x ) = x t is APN (Almost P erfect Nonlinear) o v er F 2 n for infin itely many v alues of n . Equ iv alen tly , t is exceptio nal if the binary cyclic co de of length 2 n − 1 with t wo zeros ω , ω t has minim um distance 5 for infinitely man y v alues of n . The co njecture w e pro v e states that eve ry exceptional num b er has the form 2 i + 1 or 4 i − 2 i + 1. Key wor ds: Absolutely ir reducible p olynomial, co ding th eory, cryp tograph y . Email addr esses: f. hernando @ucc.ie (F ernand o Hern ando), gary.mcg uire@ucd .ie (Gary McGuire). 1 P ostdo ctoral Researc her at the C laude Sh annon Institute, Science F oundation Ireland Gr an t 06/MI/006, and also supp orted b y MEC MTM2007-647 04 (Spain). 2 Researc h supp orted b y th e Claude Shannon Institute, Scie nce F oun dation Ireland Gran t 06/MI/006. Preprint submitted to Elsevier 11 No v em b er 2018 1 In tro duction The sequ ence of num b ers of the form 2 i + 1 or 4 i − 2 i + 1 (where i ≥ 1) is 3 , 5 , 9 , 13 , 17 , 33 , 57 , 65 , 129 , 241 , 25 7 , 513 , 993 , 102 5 , . . . . This is seque nce num b er A064386 in the On-Line Encyc lop edia of In teger Sequences . It has b een known for a lmo st 40 y ears that these num b ers are ex- c eptional n um bers, in the sense w e will define shortly . No further exceptional n um bers w ere found, a nd it w as conjectured that this sequence is the com- plete list of exceptional n um bers. In this article we complete the pr o of of this conjecture. So mewhat surprisingly , the sequence of exceptional n um b ers arises in t w o different contexts , as explained in the excellen t surv ey article o f Dillo n [2]. W e now pro ceed to giv e t hese tw o differen t motiv atio ns for the conjecture. 1.1 Co ding the ory W e fix our base field F 2 . Let w b e a primitiv e (2 n − 1)-th ro ot of unit y in an extension of F 2 , i.e., a primitiv e elemen t of F 2 n . F o r every o dd t ≥ 3, w e define C t n as t he cyclic co de ov er F 2 of length 2 n − 1 with tw o zeros w , w t . It is w ell kno wn t ha t if t = 3, the co de C 3 n has minim um distance 5 for ev ery n ≥ 3. This co de is called the 2-error-cor r ecting BCH co de. W e w an t to find other v alues of t (fixed with resp ect to n ) fo r whic h the co de C t n has minim um distance 5 for infinitely many v alues of n . Those v alues of t ha ving this prop erty are called exceptional . The only kno wn exceptional v alues fo r t are n umbers of the form t = 2 i + 1 (kno wn in the field of coding t heory as Gold n umbers) and t = 4 i − 2 i + 1 (kno wn as K a sami-W elc h n um bers). W e giv e more o n the precise history in Section 2.1. The conjecture stated b y Jan w a-McGuire-Wilson [5] is Conjecture 1: The only exc eption a l values for t ar e the Gold and Kasam i- Welch numb ers. Equiv a len tly , the conjecture sa ys that for a fixed odd t ≥ 3, t 6 = 2 i + 1 o r t 6 = 4 i − 2 i + 1, the co des C t n of length 2 n − 1 ha v e codew ords of w eigh t 4 for all but for finitely many v a lues of n . In this pap er we prov e Conjecture 1. 2 1.2 Crypto gr aphy The second appro a c h to this problem comes from cryptograph y . One of the desired prop erties for a n S-b ox used in a blo ck cipher is to ha v e the b est po s- sible resistance against differen tial at t ac ks, i.e., an y given plain text difference a = y − x pro vides a ciphertext difference f ( y ) − f ( x ) = b with small prob- abilit y . More formally , a function f : F 2 n → F 2 n is said to b e APN (Almost P erfect Nonlinear) if for ev ery a, b ∈ F 2 n with a 6 = 0 w e hav e ♯ { x ∈ F 2 n | f ( x + a ) + f ( x ) = b } ≤ 2 . Ov er a field of c haracteristic 2, APN functions provid e optimal resistance to differen tial cryptanalysis . Monomial functions f ( x ) = x t from F 2 n − → F 2 n are often conside red f o r use in applications. The exponent t is called exceptional if f ( x ) = x t is APN on infinitely man y extension fields of F 2 . The conjecture stated b y Dillon [2] is Conjecture 2: The only exc eptional ex p onents ar e the Gold and Kasami- Welch numb ers. Of course, D illon knew that Conjecture 2 is the same as Conjec ture 1, a s w e explain below. Conjecture 2 sa ys that for a fixed odd t ≥ 3 , t 6 = 2 i + 1 o r t 6 = 4 i − 2 i + 1, the function f ( x ) = x t is APN on at most a finite n um b er of fields F 2 n . In this pap er w e prov e Conjecture 2. 1.3 Summary of Pap er In Section 2 w e will explain wh y Conjecture 1 a nd Conjecture 2 a re the same. Section 3 giv es some known results and some background theory that w e will need. The pro of naturally splits in to tw o cases. W e use t he notat io n t = 2 i ℓ + 1, where i ≥ 1, and ℓ ≥ 3 is o dd. The t w o cases are dependen t on the v alue of g cd ( ℓ, 2 i − 1). In Section 4 w e recall a result of Jedlic k a, whic h pro v es the result in the case g cd ( ℓ, 2 i − 1) < ℓ . In Section 5 w e giv e a proo f of the main theorem of this pap er, Theorem 16, whic h pro v es the case g cd ( ℓ, 2 i − 1) = ℓ . In Section 6 we giv e a coun terexample to Conjecture 3 (stated b elow). 3 2 Bac kground This pro ofs in this pap er concern the absolute irreducibilit y of certain p oly- nomials. In this section w e will outline ho w these p olynomials arise from Con- jectures 1 and 2. 2.1 Co ding The ory It is w ell know n that co dew ords o f w eigh t 4 in C t n are equiv alent to the p oly- nomial f t ( x, y , z ) = x t + y t + z t + ( x + y + z ) t (1) ha ving a rational p oint ( α, β , γ ) ov er F 2 n with distinct co ordinates. No tice that x + y , x + z and y + z divide f t ( x, y , z ), so we ma y restrict ourselv es to rational po in ts of the homog eneous p olynomial g t ( x, y , z ) = f t ( x, y , z ) ( x + y )( x + z )( y + z ) . (2) Jan w a-Wilson [6] pro vide the following result using the W eil b o und. Prop osition 1 If g t ( x, y , z ) has an absolutely irr e ducible f a ctor define d over F 2 then g t ( x, y , z ) has r ational p oints ( α , β , γ ) ∈ ( F 2 n ) 3 with distinct c o or di- nates for al l n sufficiently lar ge. The follo wing conjecture w as prop osed by Jan w a-McGuire-Wilson. Conjecture 3: The p olynomial g t ( x, y , z ) is abs olutely irr e ducible for al l t n ot of the form 2 i + 1 or 4 i − 2 i + 1 . W e giv e a coun terexample (found with MA GMA) to Conjecture 3 in Sec tion 6. A sligh tly w eaker form of Conjecture 3 is: Conjecture 3 ′ : The p olynomial g t ( x, y , z ) has an absolutely irr e ducible fa ctor define d over F 2 for al l t not of the form 2 i + 1 or 4 i − 2 i + 1 . By Prop osition 1 and the discussion ab o v e, it is cle ar that Conjecture 3 ⇒ Conjecture 3 ′ ⇒ Conjecture 1. In this pap er we will pro v e Conjecture 3 ′ , and as a result, w e pro v e Conjecture 1. 4 Notice that g t ( x, y , z ) has no singular p oin ts at the infinit y , th us the latter conjecture may b e reform ulated using g t ( x, y , 1) instead of g t ( x, y , z ). W e write f t ( x, y ) for f t ( x, y , 1), a nd we write g t ( x, y ) f o r g t ( x, y , 1). F or the known exce ptional v a lues of t , that is, whe n t has the form 2 i + 1 or 4 i − 2 i + 1 , the p olynomial g t ( x, y ) is kno wn to not b e absolutely irreducible, and the factorization is describ ed in [6 ]. W e also remark that for some v alues of t , such as t = 7, g t ( x, y ) is no nsingular a nd therefore absolutely irreducible, but it is false that g t ( x, y ) is no nsingular for all t not of the form 2 i + 1 or 4 i − 2 i + 1 . 2.2 Crypto gr aphy It is w ell kno wn that h t ( x, y ) = ( x + 1 ) t + x t + ( y + 1) t + y t ( x + y )( x + y + 1) . has no rationa l p oints ov er F 2 n b esides those with x = y a nd x = y + 1 if and only if x t is APN o v er F 2 n . Analogous to Prop osition 1, Jedlic k a [7] show ed that as a consequence of the W eil b ound w e hav e the follo wing result. Prop osition 2 If h t ( x, y ) has an absolutely irr e ducible factor over F 2 then h t ( x, y ) has r ational p oints ove r F 2 n b esides those with x = y and x = y + 1 for al l n sufficiently lar ge. The follo wing conjectures are essen tially stated in [7]. Conjecture 4: The p ol ynom ial h t ( x, y ) is absolutely irr e ducibl e p olynomial for al l t not of the form 2 i + 1 or 4 i − 2 i + 1 . A slightly we ak er v ersion of this conjecture is: Conjecture 4 ′ : The p olynomial h t ( x, y ) has an a bsolutely irr e ducible fac tor define d over F 2 for al l t not of the form 2 i + 1 or 4 i − 2 i + 1 . By Prop osition 2 and the discussion ab o v e, it is cle ar that Conjecture 4 ⇒ Conjecture 4 ′ ⇒ Conjecture 2. In this pap er we will pro v e Conjecture 4 ′ , and as a result, w e pro v e Conjecture 2. W e giv e a counte rexample to Conjecture 4. 5 2.3 Putting the Pr o b lems T o gether The follo wing is w ell kno wn to researc hers in the area. Lemma 3 Conje ctur e 3 is true iff Conje c tur e 4 is true. C o nje ctur e 3 ′ is true iff Conj e ctur e 4 ′ is true. Pro of: F a ctoring o ut y t from ( x + 1 ) t + x t + ( y + 1) t + y t and letting X = x +1 y and Y = x y giv es ( x + 1) t + x t + ( y + 1) t + y t = y t [ X t + Y t + 1 + ( X + Y + 1) t ] . Therefore, w e can study the irreducibilit y of h t ( x, y ) o r that of g t ( x, y ), they are equiv alent. W e can sa y ev en more: the monomial x t is an APN function o ver F 2 n if and only if the co de C t n has minim um distance 5. This show s that Conjecture 1 is true iff Conjecture 2 is true. Conjecture 3 ⇐ ⇒ Conjec ture 4 ⇓ ⇓ Conjecture 3 ′ ⇐ ⇒ Conjecture 4 ′ ⇓ ⇓ Conjecture 1 ⇐ ⇒ Conjec ture 2 In this pap er w e will pro v e Conjecture 3 ′ , see the t a ble b elo w. This is equiv alen t to pro ving Conjecture 4 ′ , and so implies b oth Conjectures 1 and 2. How ev er, w e giv e a coun terexample in Section 6 that sho ws that Conjectures 3 and 4 are false in g eneral. Notation: Throughout w e will let t = 2 i ℓ + 1, where i ≥ 1, a nd ℓ ≥ 3 is o dd. W e use the notation S ing ( g t ) to denote the set of all singular p oin ts of g t . The follow ing b ox summarizes kno wn results b efore this pap er, and what is done in this pap er. 6 F unction Exceptional Constrain ts Author x 2 i +1 Y es APN iff ( i, n ) = 1 Gold [4 ] x 4 i − 2 i +1 Y es APN iff ( i, n ) = 1 v an L int-Wilson [10], Jan w a-Wilson [6] x t No t ≡ 3( mod 4) , t > 3 Jan w a -McGuire-Wilson [5] x 2 i ℓ +1 No g cd ( ℓ, 2 i − 1) < ℓ Jedlic k a [7] x 2 i ℓ +1 No g cd ( ℓ, 2 i − 1) = ℓ This pap er Jan w a-McGuire-Wilson prov ed the i = 1 case. The full pro of of Conjecture 3 ′ divides into tw o cases, a ccording a s g c d ( ℓ, 2 i − 1) < ℓ or g cd ( ℓ, 2 i − 1) = ℓ . Jedlic k a prov ed the case g cd ( ℓ, 2 i − 1) < ℓ . In the presen t w o rk w e giv e a proo f of Conjecture 3 ′ in t he remaining case when g cd ( ℓ, 2 i − 1) = ℓ . W e sho w that Conjecture 3 is f alse in general. This completes the classification of exceptional exp o nen ts. 3 Singularities and Bezout’s Theorem Consider P = ( α, β ), a p oin t in the plane. W rite f t ( x + α , y + β ) = F 0 + F 1 + F 2 + F 3 + · · · where F m is homo g eneous of degree m . The m ultiplicit y of f t at P is the smallest m with F m 6 = 0, and is denoted by m P ( f t ). In this case, F m is called the ta ngen t cone. Recall t he notation that t = 2 i ℓ + 1, where i ≥ 1, a nd ℓ ≥ 3 is o dd. W e let λ = α + β + 1, then straigh tforw ard calculations [6] giv e F 0 = α t + β t + λ t + 1 F 1 = ( α t − 1 + λ t − 1 ) x + ( β t − 1 + λ t − 1 ) y F 2 i = ( α t − 2 i + λ t − 2 i ) x 2 i + ( β t − 2 i + λ t − 2 i ) y 2 i F 2 i +1 = ( α t − 2 i − 1 + λ t − 2 i − 1 ) x 2 i +1 + ( β t − 2 i − 1 + λ t − 2 i − 1 ) y 2 i +1 + λ t − 2 i − 1 ( x 2 i y + xy 2 i ) and F j = 0 for 1 < j < 2 i . A p oin t P = ( α , β ) is singular if and only if F 0 = F 1 = 0, whic h happens if and only if α, β and λ are ℓ -th ro ots of unity (see [6]). W e distinguish three types of singular p oint. 7 (I) α = β = λ = 1. (I I) Either α = 1 and β 6 = 1, or β = 1 and α 6 = 1, or α = β 6 = 1 and λ = 1. W e divide these singular p oints into t w o cases : (I I.A) Where I I holds and α, β ∈ GF (2 i ) (I I.B) Where I I holds and α, β not b oth in GF (2 i ). (I I I) α 6 = 1, β 6 = 1 and α 6 = β . W e divide these singular p oints into t w o cases : (I I I.A) Where I I I holds and α, β ∈ GF (2 i ) (I I I.B) Where I I I holds and α, β not b oth in GF (2 i ). No w w e summarize some prop erties already kno wn, f or more details see [5]. Lemma 4 If F 2 i 6 = 0 then F 2 i = ( Ax + B y ) 2 i wher e A 2 i = α 1 − 2 i + λ 1 − 2 i and B 2 i = β 1 − 2 i + λ 1 − 2 i . The pro of is ob vious, b ecause w e are in c haracteristic 2. The imp ortance o f this lemma is that there is only one distinct linear factor in F 2 i . Another useful fact is that the opp osite is true for F 2 i +1 , as sho wn in [5]: Lemma 5 F 2 i +1 has 2 i + 1 distinct line ar f a ctors. 3.1 Classific ation of Singularities The next step is to des crib e how many singularities of each type there are, and to find their multiplic ities. Clearly there is only o ne singularity of ty p e I. There ar e ( ℓ − 1) p oints of t yp e (1 , β ) with β ℓ = 1 and β 6 = 1. So, there are also ( ℓ − 1) of t yp e ( α , 1) and ( ℓ − 1) of ty p e ( α, α ) with α ℓ = 1 and α 6 = 1. In total there are 3( ℓ − 1) p oin ts of type I I. F or p oints of t yp e I I I there are ( ℓ − 1 ) ch oices for α 6 = 1, and th us there are ( ℓ − 2) c hoices for β with β 6 = 1 and β 6 = α . Ho w ev er, not all these c hoices lead to a v alid singular p o in t. W e upp er b ound the n um ber o f v a lid ch oices in the next lemma. Lemma 6 F or every α with α ℓ = 1 and α 6 = 1 ther e ex i sts a β with β ℓ = 1 , β 6 = α and β 6 = 1 such that ( α + β + 1) ℓ 6 = 1 . Pro of: Supp ose the statemen t is false, and fix an α 6 = 1 suc h that for all β with β ℓ = 1 w e also ha v e ( α + β + 1) ℓ = 1. Let H b e { a | a ℓ = 1 } , t he set of ℓ -th ro ots o f unity . Consider the map, φ : H → H, φ ( β ) = α + β + 1 . 8 The k ey point is that this map has no fixed p oin ts. F or, if φ ( β ) = β , then α = 1, whic h is not true b y assumption. Th us φ is a p ermutation of H which is a pro duct of transp ositions of the form ( β , 1 + α + β ). Therefore φ m ust p erm ute an even num b er of p oin ts, whic h contradicts the fact that ℓ is o dd. F rom this lemma if follow s that, give n α , there are at most ( ℓ − 3) possible c hoices for β . W e can not guar a n tee that eac h of these is v alid, so we can only upp er b ound the p oin ts of t ype I I I by ≤ ( ℓ − 1)( ℓ − 3) . There are cases when this b ound is tight. The next Lemma helps us determine when m P ( f t ) is equal to 2 i and when it is 2 i + 1 . Lemma 7 L et P = ( α , β ) b e a singular p oint of f t , then F 2 i = 0 if and only if one o f the fol lowing hol d s. (1) P is of T yp e I (2) P is of T yp e II.A (3) P is of T yp e III.A (4) P is of T yp e III.B and α/β and β /λ ∈ GF (2 i ) . In this c ase, we have 1 < g cd ( ℓ , 2 i − 1) < ℓ . Pro of: W e ha ve to c hec k when α t − 2 i + λ t − 2 i = 0. Substituting t = 2 i ℓ + 1 in the fo rm ula we get α 1 − 2 i = λ 1 − 2 i , or α 2 i − 1 = λ 2 i − 1 . No w reasoning with β t − 2 i + λ t − 2 i = 0 we also o btain that either β 2 i − 1 = λ 2 i − 1 . So F 2 i = 0 if and o nly if α 2 i − 1 = β 2 i − 1 = λ 2 i − 1 . Consequen tly , F 2 i = 0 if and only if ( α/β ) 2 i − 1 = ( β /λ ) 2 i − 1 . If P is of T yp e I or I I.A or II I.A, then in fact α 2 i − 1 = β 2 i − 1 = λ 2 i − 1 = 1 . If P is o f T yp e I I.B then F 2 i 6 = 0 b ecause certainly one co efficien t do es not v anish. Finally , supp o se P is of T yp e I I I.B, and then we ma y deduce that α = C β and β = D λ for some C , D ∈ GF (2 i ). Raising to the ℓ - th p o w er yields that C , D are ℓ -t h ro ots of unit y . Letting d = g cd ( ℓ, 2 i − 1), then C, D are d -th ro ots of unit y . Because C and D cannot b e 1, w e mus t ha v e d > 1. If d = ℓ then all ℓ -th ro ots of unity are in GF (2 i ). Because P is of T yp e I I I.B, at least one o f α , β is not in GF (2 i ), so d < ℓ . Note that if ℓ = 2 i − 1 then t = 2 i ℓ + 1 = 4 i − 2 i + 1 , whic h is an exceptional v alue. W e no w list the classification in a table. W e let w ( x, y ) = ( x + 1)( y + 1)( x + y ) so that f t = w g t and m P ( f t ) = m P ( g t ) + m P ( w ). The v alues of m P ( w ) are easy to w ork o ut for the v arious singular p oints P . The implications of Lemma 7 9 can b e summarized in the f o llo wing tables. g cd ( ℓ, 2 i − 1) = 1 T ype Num ber of P oin ts m P ( f t ) m P ( g t ) I 1 2 i + 1 2 i − 2 I I 3( ℓ − 1) 2 i 2 i − 1 I I I ≤ ( ℓ − 1)( ℓ − 3) 2 i 2 i In this case, the T yp e I I p oints are all of Type II.B, and the T yp e I I I p oin ts a re all of Type I I I.B. g cd ( ℓ, 2 i − 1) = ℓ T ype Num ber of P oin ts m P ( f t ) m P ( g t ) I 1 2 i + 1 2 i − 2 I I 3( ℓ − 1) 2 i + 1 2 i I I I ≤ ( ℓ − 1)( ℓ − 3) 2 i + 1 2 i + 1 In this case, the Ty p e II p oin ts are all of T ype I I.A, and the Ty p e II I p oin ts a re all of Type I I I.A. The case 1 < g cd ( ℓ, 2 i − 1) < ℓ is a mixture o f the previous tw o cases b ecause f t ( x, y ) has p oints with m ultiplicity 2 i and p oints with m ultiplicity 2 i + 1. Nev erthele ss the upp er b ounds on the numb er of po in ts still hold. 3.2 Bezout’s The or em One of the cen tral results in our w ork uses Bezout’s theorem, whic h is a classical result in algebraic geometry and app ears frequen tly in the literature [3]. Bezout’s Theorem : Let r and s b e t w o pro jectiv e plane curve s ov er a field k of degrees D 1 and D 2 resp ectiv ely hav ing no comp onents in common. Then, X P I ( P , r , s ) = D 1 D 2 . (3) The sum runs ov er all the po in ts P = ( α, β ) ∈ k × k , and b y I ( P , r, s ) w e understand the in tersection multiplicit y of the curv es r and s at the p oint P . Notice that if r or s do es not go through P , then I ( P , r , s ) = 0. Therefore, the sum in (3) r uns o v er the singular p oin ts of the pro duct r s . Using prop erties I ( P , r 1 r 2 , s ) = I ( P , r 1 , s )+ I ( P , r 2 , s ) and deg ( r 1 r 2 ) = deg ( r 1 )+ 10 deg( r 2 ) one can generalize Bezout’s Theorem to sev eral curv es f 1 , f 2 , · · · , f r : X P X 1 ≤ i 2 and ℓ | 2 i − 1 but ℓ 6 = 2 i − 1 then the fol lowing r esults hold: (1) 2 i − 1 + 1 − ℓ > 2 . (2) ℓ − 3 2 i +1 < 1 4 . Pro of: Since ℓ | 2 i − 1 but ℓ 6 = 2 i − 1, and b oth n um b ers a re o dd, w e certainly ha v e that ℓ < 2 i − 1 − 1 . Then 2 i − 1 − 1 − ℓ > 0 so 2 i − 1 + 1 − ℓ > 2, th us (1) holds. F or (2) w e hav e that ℓ < 2 i − 1 − 1 < 2 i − 1 + 3 whic h implies ℓ − 3 2 i − 1 < 1 whic h certainly implies ℓ − 3 2 i +1 < 1 4 . 5.2 A Warm-Up C ase Theorem 13 Supp ose that g t ( x, y ) is irr e ducib le over F 2 and ℓ | 2 i − 1 but ℓ 6 = 2 i − 1 . Then g t ( x, y ) c an n ot s p l i t in two factors g 1 and g 2 with deg ( g 1 ) = deg ( g 2 ) . Pro of: W e apply Bezout’s Theorem, whic h states X P ∈ S in g ( g t ) I ( P , g 1 , g 2 ) = deg ( g 1 ) deg ( g 2 ) . 12 By L emma 7 w e kno w that F 2 i = 0. Since the tangen t cones ha v e differen t lines b y Lemma 4, Corollar y 8 tells us that the left hand side is equal to P P ∈ S in g ( g t ) m P ( g 1 ) m P ( g 2 ). Using the ta ble of singularities described in Section 3 for ℓ | 2 i − 1 we get X P ∈ S in g ( g t ) m P ( g 1 ) m P ( g 2 ) ≤ (2 i − 1 − 1) 2 + 3 ( ℓ − 1)2 2 i − 2 + ( ℓ − 1)( ℓ − 3)2 i − 1 (2 i − 1 + 1 ) . (6) Since the degrees of b ot h comp onen ts are the same, the righ t hand side of Bezout’s Theorem is exactly , (2 i − 1 ℓ − 1) 2 = 2 2 i − 2 ℓ 2 − 2 ℓ 2 i − 1 + 1 . (7) Let us compare (7) and (6). If (7) > (6), w e hav e won, and this happ ens if and only if, 2 2 i − 2 ( − ℓ + 1) + 2 i − 1 ( ℓ 2 − 2 ℓ + 1) < 0 (8) whic h is equiv alen t to 2 i − 1 ( ℓ − 1 ) > ( ℓ 2 − 2 ℓ + 1) = ( ℓ − 1) 2 . (9) So w e conclude that the condition for ( 7) > (6) is 2 i − 1 > ( ℓ − 1) (10) whic h is true b y Lemma 12 part (1). Remark: Not ice that this pro of fails when ℓ = 2 i − 1, as it should. The k ey idea in the previous proo f is to compare (7) and (6). In the next result w e hav e a sharp er bo und which will b e v ery useful for further results. Lemma 14 If ℓ | 2 i − 1 but ℓ 6 = 2 i − 1 , then deg( g t ) 2 > X P ∈ S in g ( g t ) m p ( g t ) 2 . Pro of: Suppo se not. Then, deg( g t ) 2 = (2 i ℓ − 2) 2 ≤ X P ∈ S in g ( g t ) m p ( g t ) 2 ≤ (2 i − 2) 2 + (3 ℓ − 3 ) 2 2 i + ( ℓ − 1)( ℓ − 3)(2 i + 1 ) 2 13 where the last inequality is obta ining using the ta ble of singularities describ ed in section 3. After rearr a ngemen t w e obtain, 0 ≤ 2 2 i + ℓ 2 2 i +1 + ℓ 2 − ℓ 2 2 i − 4 ℓ 2 i − 4 ℓ + 2 i +1 + 3 . Equiv a len tly , 0 ≤ 2 i (2( ℓ − 1) 2 ) − 2 i ( ℓ − 1)) + ( ℓ − 1)( ℓ − 3) . Dividing b y ( ℓ − 1) we get 2 i +1 (2 i − 1 − ( ℓ − 1 )) ≤ ℓ − 3 or 2 i − 1 − ( ℓ + 1) ≤ ℓ − 3 2 i +1 . Ho w ev er, by Lemma 12 we kno w that the left hand side is a p ositiv e integer and righ t hand side satisfies 0 < ℓ − 3 2 i +1 ≤ 1 / 4, a con tradiction. Remark: Aga in w e note that this pro o f fails if ℓ = 2 i − 1, as it should. 5.3 Pr o of Assuming Irr e ducibility over F 2 Next we prov e Conjecture 3 under the assumption in the title. Theorem 15 If g t ( x, y ) is irr e ducible over F 2 , and ℓ | 2 i − 1 but ℓ 6 = 2 i − 1 , then g t ( x, y ) is absolutely irr e ducible. Pro of: Supp ose that g t ( x, y ) is irreducible ov er F 2 , a nd that g ( x, y ) = f 1 · · · f r o v er some extension field o f F 2 . By Lemma 10, each f i has the same degree, whic h m ust be deg ( g t ) /r . If r is ev en then b y letting g 1 = f 1 · · · f r / 2 and g 2 = f 1+ r/ 2 · · · f r w e are done b y Theorem 13. W e may therefore assume that r is o dd (although o ur argument do es not use this, and is also v alid when r is ev en). W e apply (4) obtaining X P X 1 ≤ i (1 5) then w e hav e w on. After canceling the factors of ( r − 1) / 2 r , the inequalit y (12) > (1 5 ) is (2 i ℓ − 2) 2 > (2 i − 2) 2 + (2 i ) 2 (3 ℓ − 3) + (2 i + 1 ) 2 ( ℓ 2 − 4 ℓ + 3) whic h is true because it is exactly the same inequality as that in the pro of o f Lemma 1 4 . 5.4 Pr o of of Conje ctur e 3 ′ In this section w e will finally prov e Conjecture 3 ′ . Theorem 16 If ℓ | 2 i − 1 but ℓ 6 = 2 i − 1 , then g t ( x, y ) always ha s an absolutely irr e ducible factor over F 2 . Pro of: Suppose g t = f 1 · · · f r is the factorization in to irr educible factors ov er F 2 . Let f k = f k , 1 · · · f k ,n k b e the factor izat io n of f k in to n k absolutely irre- ducible factors. Each f k ,j has degree deg ( f k ) /n k , by Lemma 10. 15 Let us pro v e a n auxiliary result. Lemma 17 A l l F 2 -irr e ducible c omp onents f k ( x, y ) of g t ( x, y ) satisfy the fol- lowing c onditions: • deg( f k ) 2 ≤ X P ∈ S in g ( g t ) m P ( f k ) 2 . (16) • X 1 ≤ i
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment