Enumeration of algebraic and tropical singular hypersurfaces

We develop a version of Mikhalkin’s lattice path algorithm for projective hypersurfaces arbitrary degree and dimension, which enumerates singular tropical passing through appropriate configuration points. By proving correspondence theorem combined with the algorithm, we construct ? \delta dimensional linear space alttext="d"> d encoding="application/x-tex">d real containing alttext="StartFraction 1 Over delta factorial EndFraction left-parenthesis gamma Subscript n Baseline d Superscript right-parenthesis plus upper O minus right-parenthesis"> 1 ! ( ? encoding="application/x-tex">\frac {1}{\delta !}(\gamma _nd^n)^{\delta }+O(d^{n\delta -1}) nodes, where alttext="gamma n"> encoding="application/x-tex">\gamma _n are positive given by recursive formula. This is asymptotically comparable to number ( ) !} \left ( (n+1)(d-1)^n \right )^{\delta }+O\left (d^{n(\delta -1)} ) complex having nodes in space. In case alttext="delta equals 1"> = encoding="application/x-tex">\delta =1 give slightly better leading term.