Quaternary binary trilinear forms of norm unity/uhiknormiga kvaternaarsetest binaarsetest kolmlineaarsetest vormidest

So far trilinear forms have mostly been considered in low dimensions, in particular the dimension two (binary) case, and when the ring of scalars K is either the real numbers R or the complex ones C. The main aim in both situations has been to decide when a normalized form has norm unity. Here we consider the case of quaternions, K = H. This note is rather preliminary, and somewhat experimental, where the computer program Mathematica plays a certain role. A preliminary result obtained is that the form has norm unity if and only if the discriminant of a certain 5-dimensional quadratic form has all its principal minors nonnegative. We found also a rather unexpected similarity between the noncommutative case of H and the commutative one of R and C. Key words: trilinear form, quaternion, discriminant, principal minor. Siiani on enamik trilineaarsete vormidega toodest olnud suunatud madala dimensiooniga vormidele, eriti kahedimensionaalsele ehk binaarsele juhule. Skalaaride ring K on siin kas reaalsed numbrid R voi kompleksnumbrid C. Pohieesmargiks on molemas situatsioonis otsustada, millal normaliseeritud vormi norm on uhik. On vaadeldud kvaternioonide juhust K = H. Artiklis esitatu on usna ajutine ja osalt eksperimentaalne. Puudub taiuslik lopptulemus, kuid ajutiseks tulemuseks on see, et vorm on uhiknormiga parajasti siis, kui teatud 5-dimensionaalse ruutvormi diskriminandi koik peamiinorid osutuvad mittenegatiivseteks. On avastatud ootamatu seos mittekommutatiivse juhu H ja kommutatiivse juhu vahel. Viimasel juhul on skalaaride ring kas R voi C.