He studied at La Sapienza where he was taught by many leading scientists, for example Francesco Severi, Guido Castelnuovo, Federigo Enriques, Enrico Bompiani, Tullio Levi-Civita, Mauro Picone and the mathematical physicist Antonio Signorini (1888-1963). He was advised by Francesco Severi and published his first mathematics paper in 1931, namely Sulle aree delimitate da linee cicloidali . In this paper he studied a plane, closed, convex curve which rolls without sliding on a fixed curve in the same plane. While an undergraduate, he received the Cotronei Award for the best scholar of his year. In 1933 Martinelli was awarded his laurea for his thesis Sulle funzioni poligene di una e di due variabili complesse . He was awarded several prizes following the award of his laurea, namely the Beltrami Foundation Award, and the prestigious Fubini and Torelli Prizes which he shared with Pietro Buzano (1911-1993). Buzano had been awarded his laurea from the University of Turin in 1931.
Following the award of his laurea, Martinelli was appointed as an assistant to the chair of Mathematical Analysis, held by Francesco Severi. He moved next to be an assistant to the chair of Geometry, held by Enrico Bompiani. Between 1936 and 1938 he published five papers, namely Sugli insiemi bidimensionali di punti dello spazio fra loro omografici (1936), Sui coni proiettanti da un punto di una superficie di Jordan i rimanenti punti (1936), La formula di Cauchy per le funzioni analitiche di due variabili complesse (1937), Sulle funzioni poligene di due variabili complesse (1937) and Alcuni teoremi integrali per le funzioni analitiche di più variabili complesse (1938).
In 1939 he was promoted lecturer when he became a "libera docenza" in Mathematical Analysis. We have translated "libera docenza" as lecturer but it was similar to the German "docent" allowing the person to teach without being supervised by a professor. At this time Martinelli delivered courses on Analytical Geometry, Algebraic Geometry and Topology. Also in 1939 he joined the National Institute of Higher Mathematics, which had been founded by Severi, and he undertook research at the Institute from 1939 to 1946. He attended the Second Congress of the Italian Mathematical Union in Bologna from 4 to 6 April 1940 where he delivered the lecture Intorno alla teoria delle funzioni biarmoniche e delle funzioni analitiche di due variabili complesse . In 1943 Rudolf Fueter invited Martinelli to Zürich to give lectures on his recent research. This had been a difficult time for Italy with a Fascist government led by Mussolini which enacted racial laws between 1938 and 1943. These were aimed at discriminating against Italian Jews and Martinelli reported that Francesco Severi personally intervened to deny his Jewish colleagues access to the University of Rome's mathematics library after the racial laws went into effect.
In 1946 Martinelli entered the competition for the chair of Geometria analitica con elementi di Geometria Proiettiva e Geometria Descrittiva con Disegno at the University of Genoa. Giovanni Dantoni (1910-2005) and Guido Zappa (1915-2015) also competed for this chair and were placed second and third respectively. Dantoni did not have long to wait for a chair since he was appointed to the chair of geometry at Pisa in 1947 and then in 1954 became a professor at the University of Catania. Zappa was also appointed to a chair in 1947, the chair of analytical geometry at the University of Naples. He later moved to the chair of algebra at Florence. The year 1946 was important for Martinelli in another way for, in that year, he married Luigia Panella in Rome. She was also a mathematician and later became an associate professor in the Faculty of Engineering at La Sapienza University of Rome. They had two children, a son Roberto Martinelli and a daughter Maria Renata Martinelli born in Genoa in 1949. Maria Renata Martinelli became a mathematician, studying at La Sapienza University of Rome receiving her laurea with the thesis Interpolazione e approssimazione mediante spline functions .
Having been ranked first in the competition for the chair at Genoa, Martinelli took up the appointment in 1947. He remained at Genoa until 1954 and during these years he taught courses on Mathematical Analysis, Function Theory and Differential Geometry. In 1954 he returned to Rome when he was appointed to the chair of Geometry in the G Castelnuovo Mathematical Institute, being director of this Institute in 1968-69. He continued to hold this chair until he retired in 1984 but he continued to serve for another two years until he formally retired and was made professor emeritus in 1986. Giuseppe Tomassini (born in Rome in 1938) was one of Martinelli's students at La Sapienza in Rome graduating with his laurea in 1962 and becoming a professor at Pisa, Ferrara and Florence. He recalls his teacher in  where he also describes his major research contributions:-
Those were the years when disciplines such as Algebra and Topology were virtually absent from university courses. Martinelli was one of the first to evoke the need for a profound renewal of teaching methods and research tools. This naturally led to the problems of his main field of interest, the Theory of holomorphic functions of several complex variables, which a few years earlier, due to the excellent efforts of the French school, had had a rapid development, thanks to the systematic use of the proper methods for such disciplines. Of his organizational activity at that time, there were the seminars held in Rome on the theory of complex spaces, the CIME courses he directed on "Complex Varieties" in Varenna in 1956, and particularly important for the geometers of my generation, that on the theme "Functions and complex varieties" held since 1963 always in Varenna. During the years Severi provided significant contributions to the theory of holomorphic functions of several variables, Martinelli was fascinated, attracted by both analytical and topological-differential aspects. His works devoted to the analytic theory later complemented those relating to more properly geometric aspects with contributions to Kähler varieties and the quaternion structure. If we limit ourselves to his most relevant scientific results, the name of Martinelli is linked to the integral representation formulas for holomorphic functions of several variables and their application to residual theory and to the characterization of the trace of a holomorphic function. The representation formulas, extending Cauchy's classical style, were proved by Martinelli in a series of works ranging from 1937 to 1955. They express the value of a holomorphic function in an open A of the complex n-dimensional space by an integral on a suitable (n + k)-dimensional cycle. They involve nuclei that, unlike the case of a single variable, are not holomorphic. The (2n-1)-dimensional formula, proved in 1938, was later found in 1941 by a pupil of Bochner during the Ph.D. thesis, and used by Bochner himself in a work on the extension of holomorphic and meromorphic functions, which appeared in 1943. The affair did not pose any priority issues. Almost certainly, given the times, the two proofs were found independently and the formula is now universally accredited in mathematical literature as Bochner-Martinelli's formula.We can appreciate Martinelli's approach towards teaching by looking at the Preface to his 1968 textbook Introduzione alla teoria dell'omologia e della coomologia. Vol. I . He writes:-
I disagree with today's prevailing style, so I have tried to highlight the intuitive ideas that are the basis of abstract developments, and to illustrate the subsequent evolution of these .... For example, in order to be able to provide simple and interesting examples, I soon adopted cellular cross-links, while postponing the justification of their use until later.Here is a list of the contents of the book:
Chapter 2: Abstract and infinite complexes.
Chapter 3: Homology of a complex. (1) Definition of homology groups. (2) A digression on Abelian groups. (3) Structure of the homology groups of a finite complex. (4) Examples. (5) Some homological properties of simplicial complexes. (6) Pseudo-manifolds. (7) Normal bases for the chains of an abstract complex of finite type.
Chapter 4: Chain maps and invariance theorems. (1) Maps and homotopies of chain complexes. (2) Invariance of the homology groups under barycentric subdivision.
Chapter 5: Cohomology groups of a complex. (1) Cohomology groups. (2) Cohomology of an abstract complex with coefficients the integers, or a field. (3) Homology and cohomology of a chain complex.
... the book is a reasonably well organized introduction to homology theory. In fact, the writing seems very clear. There are many well thought out examples and all in all, this seems a very good book for a quarter or semester level course in algebraic topology. Of special value here is the author's careful exposition in Chapter 3, Section 2, of the main relevant results on abelian groups.The article  was written by Martinelli's doctoral student Giovanni Battista Rizza at the time when Martinelli retired. Rizza (born in Piazza Armerina in Sicily in 1924) studied for a doctorate at Genoa advised by Martinelli. Rizza spent almost all his career at the University of Parma where he held the chair of Higher Geometry from 1979. In the article  he explains how helpful Martinelli was in giving up his time to help and give advice whenever it was sought. He recalls that, being unable to meet during the week, they had a weekly meeting at Martinelli's home every Sunday afternoon. On one occasion, explains Rizza, Martinelli spent two hours teaching him Cartan's theory of exterior forms.
Martinelli received many honours for his outstanding contributions. One of the honours he received early in his career was the Prize for Mathematical Sciences from the Italian National Ministry of Education in 1943. This award was given after the possible candidates had been considered by the committee consisting of Francesco Severi (president), Ugo Amaldi and Antonio Signorini (1888-1963). Signorini was a mathematical physicist, best known for his work on elasticity, who had taught Martinelli when he was an undergraduate. In 1948, while in Genoa, Martinelli was elected to the Accademia Ligure di Scienze e Lettere, an Academy founded in 1798 and based in Genoa. He was elected to the Accademia dei Lincei in 1961 and to the Accademia delle Scienze di Torino in 1980.
In addition to his roles in universities, Martinelli had an important role in the national Italian mathematics scene, for example he played an important role in the Italian Mathematical Union serving for several years on its Scientific Commission. He also served on the Board of Directors of the journal Annali di Matematica.
Giuseppe Tomassini writes :-
During his life, his qualities as a gentleman and his gentleness were never diminished, and these characterised him in the years when I had him as a professor.As a final tribute to Martinelli we end with the final sentence from Giovanni Battista Rizza :-
A man of rare intellectual honesty, a generous teacher who was always available, leaves in his numerous students, direct and indirect, and more generally in all those who have known him, a memory destined to last in time.
Article by: J J O'Connor and E F Robertson