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Author: Dr. Eng. TRPE GRUEVSKI

91000 Skopje, Macedonia

street: "Nikola Rusinski" 12/2-6

phone no: (..389 91) 361 412

Topic: A L G O R I T H M S for solving polynomial algebraic equation with indefinite degree



Represents 22 years of scientific - research labour, which deals with algebraic problems, numerical mathematics, algorithms.

The author is born 1932 in village Hiciforovo, R.Macedonia. Graduated at the University in Sarajevo, 1962. Post graduation and doctor of science at the Karls’ University in Praha, 1970 and 1980. The author has published over 124 scientific works, 9 books and 24 innovations. Participated in many International Scientific conferences, internationals symposiums and Conventions. Now, he participate in the International Organization of experts in UNIDO.

In the book, consisted of 408 pages, the basis of the new Theory of constant formalism of the polynomial are obtained, beginning with forming the triangle or as it’s known Macedo-operator, which are discovered and put in natural numbers appropriate for each degree of polynomial separately. By the help of this operator every polynomial of n - degree is mutually converted in subordinated n - entry of numbers, so called nucleus, by which in further solution of the tasks, only the formalization of the work is performed. Later, the values P(m), where the P is the given polynomial, and m Z , (later also m Q, m R) are founded quickly, especially if P(1),..., P(m) should be successively calculated. This procedure enables quick calculation of integralnumerous zeros and presents innovation in mathematics. The whole calculation, i.e. formalization of calculation is performed by three mathematical operations only: adding, minusing, multiplying, no matter which degree of equation is accounted.

By that is not all. The algebraic roots of the equations with low degrees are easy for calculation. Namely, the equation roots are calculated through the calculated nucleus values. Besides, if one root is calculated, according to the classical theory it is necessary for polynomial to be divided with x-a and to reach for zeros of the quotient which is polynomial from (n - 1) degree. According to the presented new theory, such division is not necessary at all and the problem is dealt elegantly. With combination of presented facts solving the polynomial equation of n - degree in more alternatives (minimum two, maximum seven alternatives) is able, which is not recorded in the mathematics up to day. The writer presents that on many examples, up to degree 10, with closure of page no. 336.

On the bases of the new theory, another new two simplified squared equations are created. But according to the new theory, the squared equations are solved only with mathematical operation - adding, minimum two alternatives, which presents big simplification and innovation, as well as big relief to students.

According to the new theory, also a new cube equation in radicals is created, which is opposite to the Cardan’s cube equations, which is rewritten from one book to another for centuries.

The very significant is the fact that until now days, there are no theories, theorems, procedures of algorithms for solution of algebraic equation with the equation degree n > 4.

According to the new theory all algebraic equations from the higher degrees can be solved with:

1. Algorithm for calculation of the real and complex roots of algebra equation, and

2. Algorithm for numerical calculation of the real roots of algebraic equation, considered as a big innovation in mathematics.

The above algorithm are complete innovation in the polynomial algebra, by which all unsolved problems are solved, as well as, solution of many characteristical problems by determination of parameters, so that certain conditions to be fulfilled in advance, and the same are given by famous scientific workers, professors, academicians, and the same are presented in the book.

Despite up to now opinions of the mathematicians that the zeros of the polynomial of n - degree will never be calculated, doesn’t necessarily mean their "disappointment".

As it is known the 10th Hillberts’ problem is attracting the attention of many well known mathematicians in the world, but anyway in general case (when the equation with two or more unknown is given) the wanted algorithm is founded yet. 1969 year, the Russian mathematician Ju. M. Matijasevic proved that such an algorithm will be never found ( in the future). Unfortunately, claim is not true !

From the previous presented theory point of view, with theoretically derived access and by proving of their theoretical validity, the presented theory represents one rounded and unique whole.

In chapter 29 the writer in detail presents the 10 -th Hilbert problem, and then short procedure for searching wholenumerical roots of given polynomial with wholenumerical coefficient is presented, involving notion "Hilbertian", according to the famous German mathematician David Hilbert ( 1862-1943). In addition only one example for that is given, allowing the reader to think for further extension and estimation of the procedure validity.

The diofant equation are presented in chapter 30, and examples for solution of polynomial equations in larger amount of scientific areas are presented in chapter 31 , such as: theory of stability, genetic engineering, medical sciences, chemistry, robotics, accounting technique and etc.

The author’s script represents round integrity by which the problems from the polynomial algebra area are treated adequately. For introduction of the proposed text and its implementation, it is sufficient to be aware of the basic notions of the polynomial algebra and that’s why the whole mathematics apparatus is part of the contents of the text.



- The most important component of this script is originality of searching the wholenumerical roots of the polynomial. Even though according to the existing theory, it is partially solved problem, it is interesting here that in the theory itself, interesting and simple algorithm are implemented. Here is also the formation of the triangle transformation itself, called Macedo - operator and the further formalisation of the procedure. We will hereby mention the interesting table on page 373, also.

- Once, one (wholenumerical) root is calculated, the polynomial equation with lower degree still remains to be solved. It is interesting that the decreasing is extremely simple according to the presented theory, which is also easy and interesting for the reader.

It is necessary to sat that the above mentioned problems, tortured the mathematicians throughout the world for 17 centuries. So, it is not oddly that despite the existence of 230 types of mathematical disciplines, almost 65 % from the total amount of mathematician are working on the problems of algebrazation in the world today.

The reasons for such interest are many !

Namely, there isn’t such scientific area, where ALGORITHMS are not asked for solution of algebraic equations and at the same time not to have applied meaning, not only for various areas of mathematics, but also for the industrial application in: robotics, cybernetics, electronics, ballistics, in theoretical physics, in the micro and macro cosmos theory, and especially for:

- determination of stability and quality of the regulation circles and systems;

- for genetical engineering development, especially about the detail decode of the DNA secrets, the flow of miraculous molecule, which manages with the mechanism of the cells of the inheritance;

- for medical sciences development, through building up new diagnostical technique for understanding in structure and functioning of organisms;

- for chemistry development, for solution the equation in the Hickel variant of the molecule - orbital theory, which is reduced to determination of zeros of the certain polynomial;

- for accounting technique development. Despite the fact that the computers of the fifth generation are already made as prototypes, it won’t be for long that such computers will begin to be used in everyday life.

Then, the computers of the sixth generation are approaching, which will be able to deal with the mathematics logic, and then the computers of the seventh generation will arrive which would "recognize" pictures, things, as well as adoption and procession not only of " common" knowledge, but also presentation and notions.

Generally speaking, the presented theory is rounded and unique whole from the algebraisation area.

To mention that the book’s script has been already reviewed skillfully and literary by eminent experts - mathematicians from the University " St. CYRIL AND METHODIUS" - Skopje.

The book is aimed for wide variety of readers: high school students, university students, engineers, M.A., scientists, as well as for those interested in mathematics and deal with it.

The book is declared and protected in WIPO (World Intellectual Property Organization) in Geneva, as well as the Copyright Agency in R. Macedonia.

The proves for new algorithms are represented on the I Congress of mathematicians and computer specials in Macedonia, maintained in Ohrid, on 3-5 October 1996.

The author is asking for help and cooperation of individuals and institutions that are interested. For publishing this book in different languages and for further cooperation is needed aplicative activity.

The publisher is Society of mathematicians in Skopje.

Best regards,


Skopje, 07. 11. 1996 year Dr.Eng. TRPE GRUEVSKI, Ph.Dr











Prestavuva 22 godi{en nau~no - istra`uva~ki trud na avtorot, koj gi obrabotuva problemite na algebrata, numeri~kata matematika, algoritmite.

Avtorot e roden 1932 godina vo selo Ni}iforovo, Republika Makedonija. Diplomiral na Univerzitetot vo Saraevo 1962 godina. Magistriral i doktoriral na Karloviot Univerzitet vo Praga, 1970 i 1980 godina. Ima objveno preku 124 nau~no - stru~ni trudovi, 9 knigi i 24 pronajdoka. U~estvuval na brojni Me|unarodno nau~ni konferencii, sovetuvawa i simpoziumi. Sega, se nao|a na me|unarodnata organizacija na eksperti na UNIDO.

Vo knigata ima 408 stranici, se iznesuvaat osnovite na novata Teorija na neprekinat formalizam na polinomite, po~nuvaj}i so formiraweto na troagolnata transformacija ili t.n. Makedo-ooperator vo koj se pronajdeni i vneseni prirodnite broevi koi odgovaraat za sekoj stepen na polinomot posebno. So pomo{ta na ovoj operator sekoj polinom od n-ti stepen vzaemno ednozna~no se preslikuva vo podredena n-torka na broevi, nare~eno jadro, so koe vo prodol`enie na re{avaweto na zada~ite se vr{i samo formalizacija na rabotite. Potoa dosta brzo se nao|aat vrednostite P (m), kade P e dadeniot polinom, a m Z (a podocna i m Q, m R ), posebno ako sukcesivno treb da se dobijat P (1),..., P(m). Ovaa postapka ovozmo`uva brzo nao|awe na celobrojnite nuli i pretstavuva novina vo matematikata. Celokupnata presmetka odnosno formalizacija na presmetkata se vr{i samo so tri matemati~ki operacii: sobirawe, odzemawe i mno`ewe, nezavisno od toa za koj stepen na ravenkata se raboti.

No toa sekako ne e se. Pritoa lesno se nao|aat korenite na algebarskite ravenki. Imeno, se dobivaat koreni na ravenkite preku dobienite vrednosti od jadroto. Osven toa, koga }e se najde eden koren a , spored klasi~nata teorija potrebno e dadeniot polinom da se podeli so h - a i da se baraat nulite na koli~nikot {to e polinom od (n-1)-va stepen. Spored prezentiranata nova teorija, takvoto delewe ne e voop{to potrebno i problemot se nadminuva mnogu elegantno. So kombinacija na iznesenite fakti se ovozmo`uva re{evawe na polinomnite ravenki od stepen n vo pove}e varijanti ( minimum dve, a maksimum i do sedum varijanti), {to do denes ne e zabele`ano vo matematikata. Avtorot toa go prezentira na golem broj primeroci, do stepen 10, zaklu~no so 336 strana.

Na osnova novata teorija, sozdadeni se u{te dve novi poednostavni kvadratni ravenki. No spored novata teorija, kvadratnite ravenki se re{avat isklu~itelno samo so matemati~ka operacija - sobirawe, i toa vo minimum dve varijanti, {to pretstavuva isto taka golemo poednostavuvawe i inovacija kako i golemo olesnuvawe na u~enicite.

Vrz osnova na novata teorija, sozdadena e isto taka nova kubna ravenka so radikali, {to e ampandant na Kardanovata kubna ravenka, koja so vekovi se prepi{uva od kniga vo kniga.

No ona {to e mnogu zna~ajno e i toa, {to do denes generalizirano ne postojat teorii, teoremi, metodi, postapki ili algoritmi za re{avawe na algebarskite ravenki so stepen na ravenkata pogolem od n > 4.

Spored novata teorija isto taka site algebarski ravenki od povisokite stepeni mo`at da se re{avaat so:

1. Algoritam za presmetuvawe na realnite i kompleksnite koreni na algebarski ravenki, i

2. Algoritam za numeri~ko presmetuvawe na realnite koreni na algebarskite ravenki, {to se smeta za golem prodor vo matematikata.

Gornite algoritmi se potpolna novina vo oblasta na algebrata na polinomite, so koi se re{avaat site do denes nere{eni problemi kako i re{avawe na niza karakteristi~ni problemi so opredeluvawe na parametri, taka da odredeni uslovi odnapred bidat zadovoleni, a istata se zadavani od poznati nau~ni rabotnici, profesoti i akademici i istite se prezentirani vo trudot.

I pokraj dosega{nite tvrdewa od strana na matemati~arite deka nulata na polinomot od n-ti stepen nikoga{ nema da bidat pronajdeni ne treba da zna~i i nivno "razo~aruvawe".

Od gledna to~ka na prethodno izlo`enata teorija, so teoretski izveden pristap i so doka`uvawe na nivnata teoretska validnost, prezentiranata teorija pretstavuva edna zaokru`en i edinstvena celina.

Vo glava 29 avtorot detalno go prezentira 10 - tiot Hilbertov problem, a potoa se dava skratena postapka za prebaruvawe na celobrojnite koreni na daden polinom so celobrojni koeficienti, voveduvaj}i pritoa poim "Hilbertijan", spored poznatiot germanski matemati~at David Hilbert (1862-1943). Vo prilog na toa daden e samo eden primer, ostavaj}i prostor na ~itatelot da razmisluva za ntamo{no pro{iruvawe i oceni validnosta na postapkata.

Kako {to e poznato 10-tiot Hilbertov problem go privlekuvalo vnimanieto na mnogu istaknati matemati~ari vo svetot, no sepak vo op{t slu~aj (koga e dadena ravenkata so dve ili pove}e nepoznati) baraniot algoritam denes ne e pronajden. 1969 godina, Ruskiot matemati~ar M.Matija{evi} doka`al deka takov algoritam nikoga{ nema ni da se pronajde ( vo idnina ). Za `al, tvrdeweto ne e to~no !

Vo 30 glava razgledani se Diofatnovite ravenki, a vo glava 31 izneseni se primeri za re{avawe na polinomnite ravenki vo pogolem broj nau~ni disciplini, kako {to se: teorija na stabilnost, geneti~kiot in`inering, medicinskite nauki, hemijata, robotikata, smeta~kata tehnika i drugi.

Rakopisot na avtorot pretstavuva zaokru`ena celina so koja na adekvaten na~in se tretiraat problemite od oblasta na algebrata na polinomite. Za zapoznavawe na predlo`eniot tekst i negovata primena dovolno e da se poznavaat osnovnite poimi od algebrata na polinomite i zatoa celiot matemati~ki aparat e sostaven del na sord`inata na tekstot.

- Najva`na komponenta na ovoj rakopis e sekako orginalnosta na prebaruvaweto na celobrojnite koreni na polinomite. Iako vrz osnova na postoe~kata teorija toa e delimi~no re{liv problem, ovde e interesno {to vo samata teorija se vgradeni interesi i ednostavni algoritmi. Tuka e i samoto formirawe na triagolnata transformacija, nare~ena Makedo-operator i samata natamo{na formalizacija na postapkata. Tuka bi ja spomenale i interesnata tabela na strana 373.

- Koga ve}e e najden eden (celobroen ) koren, ostanuva da se re{i polinomnata ravenka so sni`en stepen. Interesno e {to sni`uvaweto e krajno ednostavno spored izlo`enata teorija i {to e navistina lesno i interesno za ~itatelot.

Potrebno e da se napomene deka gore navedenite problemi matemati~arite vo svetot gi ma~elo celi 17 veka. Pa zatoa ne e ni ~udno {to i pokraj postoeweto na 230 vidovi matemati~ki disciplini, denes vo svetot rabotat 65% od vkupniot broj na matemati~ari na problemite vo algebraizacijata.

Pri~inite za takviot interes se pove}ekratni !

Imeno, skoro da nema nau~na oblast, kade ne se biraat ALGORITMI za re{avawe na algebarskite ravenki a da pritoa aplikativno zna~ewe, ne samo za razni oblasti od matematikata, tuku za {irokite industriski aplikacii vo: robotikata, kibernetikata, elektronikata, balistikata, vo teoretska fizika, vo toerijata na makro i mikro kosmosot, a pogotovo za:

- odreduvawe na stabilnosta i kvalitetot na regulacionite kola i sistemi;

- za razvojot na geneti~kiot in`iwering, pogotovo okolu detalnoto de{ifrirawe na tajnite na DNK, tekot na ~udesniot molekul, koj upravuva so mehnizmot na kletkite na nasledstvoto;

- za razvojot na medicinskite nauki, preku izgradba na nova dijagnosti~ka tehnika, za uvid vo strukturata i funkcioniraweto na organizmite;

- za razvojot na hemijata, za re{avawe na ravenkite vo Hikelovata varijanta na molekulsko-orbitalnata teorija, koja se sveduva na nulite na odredenite polinomi;

- za razvojot na smeta~kata tehnika. I pokraj faktot {to smeta~ite na pettata generacija se ve}e izraboteni kao prototipovi, nema da mine dolg period, a smeta~ite na pettata generacija }e po~nat da se koristat vo sekojdnevniot `ivot.

Potoa doa|aat na red smeta~ite od {estata generacija, koj }e mo`at da ja sovladuvaat matemati~kata logika, a potoa na red }e dojdat smeta~ite od sedmata generacija koja }e mo`e "da gi raspoznava" slikite, predmetite, kako i }e prifa}aat i obrabotuvaat ne samo " voobi~eno" znaewe tuku i prestavi i poimi.

Genralno re~eno, prezentiranata teorija prestavuva edna zaokru`ena i edinstvena celina, od oblasta na algebraizacijata. koja pokraj ostanatoto ima ne samo teoretsko, nau~no - edukativno tuku pogotovo prakti~no zna~ewe, bidej}i se raboti za originalen (unikaten) nau~en trud.

Trudot na d-r Gruevski vo se se vklopuva spored teoretskite postulati na slavniot Ajn{tajn, koj vika: Sekoja opstojna teorija mora da poseduva vnatre{no sovr{enstvo i nadvore{no opravduvawe, {to bez somnenie ovoj trud toa i go poseduva.

Napomenuvame deka rakopisot na knigata e ve}e recenziran stru~no i jazi~no od strana na eminentni stru~waci matemati~ari od Univerzitetot "Sv. KIRIL I METODIJ" od Skopje. Knigata e nameneta za {irok krug ~itateli: gimnazijalci, studenti, in`iweri, magistri, doktori na nauki, nau~ni institucii, kako i na lu|eto od univerzitet.

Knigata e prijavena i za{titena vo WIPO ( World Intellectual Property Organization ) vo @eneva. kako i vo Avtorskata Agencija na Republika Makedonija. Dokazite za novite algoritmi prezentirani se na 1-viot Kongres na matemati~ari i informati~ari na Makedonija, odr`an vo Ohrid, od 3-5 Oktomvri 1996 godina.

Avtorot bara pomo{ i sorabotka na zainteresirani ppoedinci i institucii za publikuvawe na knigata na drugi jazici i za pontamo{na rabota i aplikativna dejnost.

Izdava~ e dru{tvoto na matemati~ari na grad Skopje, R.E. "NUMERUS".


Skopje 25.11.1996 god.


D-r in`. Trpe Gruevski


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