|
|
| Line 1: |
Line 1: |
| − | ==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a1202501.png" />-arcs in projective planes.==
| + | "'''Arc'''" may refer to: |
| − | A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a1202504.png" />-arc in the Desarguesian [[Projective plane|projective plane]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a1202505.png" /> over the [[Galois field|Galois field]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a1202506.png" /> is a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a1202507.png" /> points, no three of which are collinear. It is immediate that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a1202508.png" />, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a1202509.png" /> is odd, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025010.png" />. The classical example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025011.png" />-arc is a conic, that is, a set of points projectively equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025013.png" /> is even, then all the tangents to the conic pass through a common point, called the nucleus; hence the set of points of a conic together with the nucleus is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025014.png" />-arc. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025015.png" />-arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025016.png" /> is called an oval and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025017.png" />-arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025019.png" /> even, is called a hyperoval (cf. also [[Oval|Oval]]).
| |
| | | | |
| − | In his celebrated 1955 theorem, B. Segre proved that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025021.png" /> odd, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025022.png" />-arc is a conic. This important result, linking combinatorial and algebraic properties of sets of points, was of great importance in the early development of the field of finite geometry, and many results in the same spirit have been proved.
| + | * a topological space homeomorphic to the unit interval; see [[arc (topology)]]; |
| − | | + | * a set of points in a finite projective space in general position; see [[arc (projective geometry)]]. |
| − | The situation when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025023.png" /> is even is quite different. Apart from the "classical" examples provided by a conic together with its nucleus mentioned above, there are currently (1998) seven infinite families of hyperovals known, and several examples which do not at present fit into any known infinite family. The classification of hyperovals is known only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025024.png" />; the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025025.png" /> still relies on a computer search.
| |
| − | | |
| − | ==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025026.png" />-arcs in projective space.==
| |
| − | A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025029.png" />-arc in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025030.png" />-dimensional [[Projective space|projective space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025031.png" /> is a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025032.png" /> points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025033.png" /> and at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025034.png" /> in each hyperplane. (This definition coincides with the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025035.png" />-arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025036.png" /> given above.) Further, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025037.png" />-arc of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025040.png" />, exists if and only if a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025041.png" />-arc of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025042.png" /> exists. A linear maximum distance separable code is a linear code of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025043.png" />, dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025044.png" /> and minimum distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025045.png" /> (cf. also [[Coding and decoding|Coding and decoding]]). It is well-known that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025046.png" /> these notions are equivalent; as each can be viewed as a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025047.png" /> vectors in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025048.png" />-dimensional [[Vector space|vector space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025049.png" /> with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025050.png" /> vectors being linearly independent.
| |
| − | | |
| − | The classical example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025051.png" />-arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025052.png" /> is a normal rational curve, that is, a set of points projectively equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025053.png" />. The only known (1998) non-classical examples are examples in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025054.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025055.png" /> even, constructed by L.R.A. Casse and D.G. Glynn, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025056.png" />-arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025057.png" /> constructed by Glynn. The main open (1998) problem in the area is the resolution of the so-called main conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025059.png" />-arcs and maximum distance separable codes, which is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025060.png" />, then the size of a largest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025061.png" />-arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025062.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025063.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025064.png" /> is even and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025065.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025066.png" />, and is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025067.png" /> in all other cases. The main conjecture has been settled for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025068.png" /> and in a number of further cases. It is also of great interest to characterize the largest-size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025069.png" />-arcs, and to determine the size of the second-largest complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025070.png" />-arcs in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025071.png" />, where a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025072.png" />-arc of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025073.png" /> is complete if it is contained in no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025075.png" />-arc of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025076.png" />.
| |
| − | | |
| − | ==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025077.png" />-arcs in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025078.png" />.==
| |
| − | For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025079.png" />, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025081.png" />-arc (or arc of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025083.png" />) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025084.png" /> is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025085.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025086.png" /> points such that each line meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025087.png" /> in at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025088.png" /> points and there is a line meeting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025089.png" /> in exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025090.png" /> points. It is immediate that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025091.png" /> with equality if and only if each line meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025092.png" /> in 0 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025093.png" /> points. Equality also implies that either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025094.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025095.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025096.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025097.png" />-arc with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a12025098.png" /> is called a maximal arc of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250100.png" />, and is non-trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250101.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250102.png" /> is even, there are examples of non-trivial maximal arcs of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250103.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250104.png" /> dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250105.png" /> (due to R.H.F. Denniston and J.A. Thas). On the other hand, S. Ball, A. Blokhuis and F. Mazzocca have shown that non-trivial maximal arcs in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250106.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250107.png" /> is odd, do not exist. (The proof appears in [[#References|[a2]]].)
| |
| − | | |
| − | See [[#References|[a3]]] for a survey on each topic mentioned above; for a comprehensive account including more details, results and the references, see [[#References|[a2]]], Chaps. 8, 9.
| |
| − | | |
| − | The definitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250108.png" />-arc, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120250/a120250109.png" />-arc, oval, hyperoval and maximal arc in a non–Desarguesian projective plane are the combinatorial definitions given above, but in this case there are relatively few examples and the theory is not so well-developed.
| |
| − | | |
| − | ====References====
| |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W.P. Hirschfeld, J.A. Thas, "General Galois geometries" , Oxford Univ. Press (1991)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.W.P. Hirschfeld, "Projective geometries over finite fields" , Oxford Univ. Press (1998) (Edition: Second)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.A. Thas, "Projective geometry over a finite field" F. Buekenhout (ed.) , ''Handbook of Incidence Geometry, Buildings and Foundations'' , Elsevier (1995) pp. Chap. 7; 295–348</TD></TR></table>
| |