Micronutrient deficiency Micronutrient deficiency Micronutrient deficiency or dietary deficiency is not enough of one or more of the micronutrients required for optimal plant or animal health. In humans and other animals they include both vitamin deficiencies and mineral deficiencies, whereas in plants the term refers to deficiencies of essential trace minerals. Micronutrient deficiencies affect more than two billion people of all ages in both developing and industrialized countries. They are the cause of some diseases, exacerbate others and are recognized as having an important impact on worldwide health. Important micronutrients include iodine, iron, zinc, calcium, selenium, fluorine, and vitamins A, B, B,
Trace field of a representation of formula_4 (see for example in ). The "invariant trace field" is equal to the trace field of the subgroup formula_7 generated by all squares of elements of formula_4 (a finite-index subgroup of formula_4). The invariant trace field of Fuchsian groups is stable under taking commensurable groups. This is not the case for the trace field; in particular the trace field is in general different from the invariant trace field. Let formula_4 be a Fuchsian group and formula_11 its trace field. Let formula_12 be the formula_11-subalgebra of the matrix algebra formula_14 generated by the preimages of elements of formula_4. The