In 2020 Parusiński and Rond proved that every algebraic set X ⊂ R^n is homeomorphic to an algebraic set X’ ⊂ R^n which is described globally (and also locally) by polynomial equations whose coefficients are real algebraic numbers. In general, the following problem was widely open: Open Problem. Is every real algebraic set homeomorphic to a real algebraic set defined by polynomial equations with rational coefficients? The aim of my PhD thesis is to provide classes of real algebraic sets that positively answer to above Open Problem. In Chapter 1 I introduce a new theory of real and complex algebraic geometry over subfields recently developed by Fernando and Ghiloni. In particular, the main notion to outline is the so called R|Q-regularity of points of a Q-algebraic set X ⊂ R^n. This definition suggests a natural notion of a Q-nonsingular Q-algebraic set X ⊂ R^n. The study of Q-nonsingular Q-algebraic sets is the main topic of Chapter 2. Then, in Chapter 3 I introduce Q-algebraic approximation techniques a là Akbulut-King developed in collaboration with Ghiloni and the main consequences we proved, that are, versions ‘over Q’ of the classical and the relative Nash-Tognoli theorems. Last results can be found in in Chapters 3 & 4, respectively. In particular, we obtained a positive answer to above Open Problem in the case of compact nonsingular algebraic sets. Then, after extending ‘over Q’ the Akbulut-King blowing down lemma, we are in position to give a complete positive answer to above Open Problem also in the case of compact algebraic sets with isolated singularities in Chapter 4. After algebraic Alexandroff compactification, we obtained a positive answer also in the case of non-compact algebraic sets with isolated singularities. Other related topics are investigated in Chapter 4 such as the existence of Q-nonsingular Q-algebraic models of Nash manifolds over every real closed field and an answer to the Q-algebrization problem for germs of an isolated algebraic singularity. Appendices A & B contain results on Nash approximation and an evenness criterion for the degree of global smoothings of subanalytic sets, respectively.

To αντικείμενο μελέτης της εργασίας αυτής είναι κατά μείζονα λόγο το θεώρημα Tarski-Seidenberg. Στο πρώτο κεφάλαιο μελετάμε το κίνητρο που ώθησε τον Tarski σε αυτή την έρευνα, εξιστορούμε την πορεία της ιδέας του από την ανακάλυψη μέχρι τη δημοσίευση και έπειτα προσπαθούμε να σκιαγραφήσουμε ευκρινώς τη συνολική επίδραση του θεωρήματος στα μαθηματικά και όχι μόνο. Για την ακρίβεια, αναφερόμαστε στην πληρότητα της Ευκλείδειας γεωμετρίας ως συνέπεια του θεωρήματος, στη συμβολή του θεωρήματος στην ανάπτυξη της ημιαλγεβρικής γεωμετρίας. Στο δεύτερο κεφάλαιο αποδικνύεται το εν λόγω θεώρημα, δηλαδή ότι η πρωτοβάθμια θεωρία των πραγματικώς κλειστών σωμάτων είναι πλήρης, με χρήση των θεωρημάτων Sturm και Sylvester. Στο τρίτο κεφάλαιο παρουσιάζεται μία διδακτική έρευνα με φοιτητές του τμήματος με σκοπό τη διάγνωση πιθανών γνωστικών κενών των φοιτητών σε θέματα της θεωρίας πολυωνύμων με πραγματικούς συντελεστές. / To study object of this work is a fortiori the Tarski-Seidenberg theorem. In the first chapter we study Tarski's motivation in this research, we recount the progress of the idea from ​​the discovery until the publication, and then we try to outline clearly the overall effect of the theorem in mathematics and beyond. In fact, we refer to the completeness of Euclidean geometry as a consequence of the theorem, in its contribution to the development of semialgebraic geometry. In the second chapter we prove the Tarski-Seidenberg theorem, namely that the first order theory of real closed fields is actually complete, using the Sturm and Sylvester theorems. In the third chapter we present a teaching research on students of the Department in purpose to diagnose potential knowledge gaps of the students concerning the theory of polynomials with real coefficients.